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The Great City Games, an open and free event for the public, is due to take place in Manchester on Friday.
Organisers said the event will go ahead as planned, but a decision on Sunday's Great Manchester Run "is expected in the next 24 hours".
The FA Cup final, EFL play-offs and the PGA Championship are also this week.
An eight-year-old girl was among those killed in Monday's suicide bombing at Manchester Arena, at the end of a concert by US singer Ariana Grande.
Prime Minister Theresa May said the UK terror threat level has been raised to its highest level of "critical", meaning further attacks may be imminent.
Manchester United cancelled a news conference on Tuesday, due to be held prior to their Europa League final against Ajax in Stockholm on Wednesday, and will wear black armbands for the match.
The club said: "Our thoughts are with the victims and their families at this terribly difficult time."
United's players held a minute's silence at training on Tuesday, and the club closed its megastore, museum, cafe and stadium tours to the public.
A staff event scheduled for Wednesday has been cancelled by executive vice-chairman Ed Woodward.
Manager Jose Mourinho said: "We are all very sad about the tragic events; we cannot take out of our minds and our hearts the victims and their families.
"We have a job to do and we will fly to Sweden to do that job. It is a pity we cannot fly with the happiness that we always have before a big game.
"I know, even during my short time here, that the people of Manchester will pull together as one."
Ajax manager Peter Bosz said: "What happened yesterday evening in Manchester is something we all feel in Ajax and on behalf of all of us at Ajax we express our sympathies with the victims that fell. The feeling that prevails is the final does not have the glow it should have.
"Tomorrow evening should be a football feast but because of the events in Manchester we are affected. It is horrible. My sympathies are heartfelt."
Football's European governing body Uefa announced a minute's silence will be observed prior to the final. The opening ceremony will also be considerably reduced as a mark of respect for the victims.
Aleksander Ceferin, president of Uefa, said he was "deeply saddened" and shocked that "so many innocent people lost their lives".
A Uefa statement said there was "currently no specific intelligence" to suggest Wednesday's game could be a target for further attacks.
"Uefa has been closely working with local authorities and the Swedish FA for many months and the terrorist risk had been taken into account since the very beginning of the project," it said.
"Furthermore, a number of additional security measures were implemented following the attacks in Stockholm last April."
There will be a minute's silence observed at Headingley cricket ground before England's one-day international against South Africa on Wednesday.
Both sets of players will also wear black armbands during the game.
The South Africa team have been told there will be extra police officers on duty at the ground and increased security at team hotels and practice.
There will also be a minute's silence before Saturday's Scottish Cup final between Celtic and Aberdeen at Hampden Park.
The Scottish FA's security and integrity officer, Peter McLaughlin, said: "We remain vigilant to the threat posed by global terrorism and are engaged in constant dialogue with colleagues at Police Scotland and the National Counter-Terrorism Security Office.
"This ongoing communication and intelligence-sharing is part of our operations protocol for all events at the national stadium, including the forthcoming Scottish Cup final."
A number of leading athletes are scheduled to participate at the Great City Games on Friday, while a public half marathon and 10km run are due to be staged in Manchester on Sunday.
Wembley hosts Saturday's FA Cup final between Arsenal and Chelsea, and the League Two and Championship play-off finals on Sunday and Monday respectively.
A Football Association spokesperson said: "Fan safety is of paramount importance and we have robust security measures in place at Wembley Stadium.
"In collaboration with the Metropolitan Police and the local authorities there will be an enhanced security operation for all upcoming events.
"All supporters are encouraged to arrive for events at Wembley Stadium as early as possible for security checks and to avoid any delays in entering the stadium."
The English Football League (EFL) added it "takes security issues extremely seriously" and urged supporters travelling to Wembley to "be vigilant of their surroundings at all times, stay alert and not be alarmed".
The Metropolitan Police says extra armed officers will be deployed at this weekend's major sports events in London, with a full review of the security and policing operations under way.
"Over the coming days as you go to a music venue, go shopping, travel to work or head off to the fantastic sporting events you will see more officers - including armed officers," said commander Jane Connors.
Golf's BMW PGA Championship starts at Wentworth on Thursday.
"As with any major event, security is the highest priority," said European Tour chief executive Keith Pelley. "It was before Monday night and it remains so.
"We're in constant dialogue with the police and security services. We are comfortable we will react in the right way if in fact we need to significantly increase our security."
Cricket's Champions Trophy will take place from 1-18 June at venues in Birmingham, London and Cardiff.
A statement from the International Cricket Council [ICC] read: "The ICC and ECB [England and Wales Cricket Board] place safety and security at the ICC Champions Trophy and ICC Women's World Cup this summer as the highest priority.
"We operate on advice from our tournament security directorate - in conjunction with the ECB and relevant authorities - to ensure that we have a robust safety and security plan for both tournaments.
"We will continue to work with authorities over the coming hours and days and review our security in line with the threat levels."
England one-day captain Eoin Morgan said his team had met their security advisers on Tuesday morning before Wednesday's match against South Africa.
"On behalf of the England cricket team, I'd like to offer our thoughts and prayers to everybody in Manchester affected by the tragic events," said Morgan.
"I'd also like to give our support to those in and around things and those most affected and those who helped out and continue to help out."
The domestic rugby union finishes this weekend, but the National Counter Terrorism security office has been in touch with Sale Sharks and every other Aviva Premiership club asking for details of any events planned by them over the next couple of weeks.
There will also be tighter security at horse racing's Epsom Derby on 3 June, with Surrey Police announcing firearms officers on patrol around the grounds.
Chief Superintendent Jerry Westerman said: "The Epsom Derby is a fantastic event which attracts thousands of people and spectators from around the world and I am confident that this year's festival will be no exception."
England Women's cricketer Danielle Wyatt was at the Ariana Grande concert and said: "Thank you for all messages - I'm safe. Was at the concert enjoying myself like many others - thoughts with victims & families."
Manchester United and Spain goalkeeper David de Gea tweeted: "Much rage, much pain. My condolences to the victims' family members involved in the atrocious attack to the heart of the city."
Manchester United forward Jesse Lingard said the "beautiful city" of Manchester "will stand together in this dark hour", captain Wayne Rooney said he was "devastated" by the news and winger Ashley Young said he was "absolutely shocked".
Former Manchester United defender Rio Ferdinand: "My thoughts & prayers are with all the families & friends affected by last night's attack in Manchester."
Former Lancashire and England cricketer Andrew Flintoff: "In the toughest of times the people of Manchester showing why this is such a great city, standing together in the face of such evil."
Manchester City players - including captain Vincent Kompany, goalkeeper Willy Caballero, forward Leroy Sane and defender Pablo Zabaleta - also tweeted their support for those affected.
Lucy Bronze, from City's women's team, said her "thoughts are with those affected" and urged people to "stick together".
Olympic and world 100m champion Usain Bolt tweeted: "Thoughts & prayers goes out to people of Manchester and all those who are affected."
Sporting events and venues in England are conducting major security reviews after 22 people were killed in an attack at Manchester Arena.