United are expected to release an official statement in the coming days.
Giggs, 42, had a year remaining on his contract as assistant manager but new boss Jose Mourinho intends to fill that role with long-time friend Rui Faria.
Unable to reach agreement over another role at the Premier League club, Giggs, who made a record 963 appearances for the club, has decided to move on.
Cardiff-born Giggs has passed all his coaching badges and has never made any secret of his desire to move into management.
Linked with Swansea last season, he felt he was ready to manage United, having served his apprenticeship under David Moyes and then Louis van Gaal.
But he was known to be less than impressed with how Van Gaal's exit was handled by United, when news of the Dutchman's sacking emerged moments after the club's FA Cup final success.
Giggs was also disappointed to be overlooked in favour of former Chelsea and Real Madrid boss Mourinho, whose appointment was confirmed in May.
Giggs joined the United academy on his 14th birthday, turning professional aged 17 in November 1990 and making his first-team debut against Everton on 2 March, 1991.
He won a record 13 league titles, two European Cups, four FA Cups and four League Cups to become the most decorated British player of all time.
He became a player-coach following the arrival of Moyes as manager in 2013, taking control of the first team for the final four games of the season after the Scot's dismissal.
Giggs was then appointed as assistant-manager by Van Gaal in 2014.
In the 2013-14 season, he was made a player-coach by Moyes and retired as a player that summer, aged 40, before signing a three-year deal as Van Gaal's number two, with the pair leading United to their first FA Cup win in 12 years last season.
Giggs represented one of the final links to the Ferguson era - and given his service and achievements at United, the club risk a potential PR disaster in letting him leave.
The Welshman was largely responsible for promoting exciting 18-year-old England striker Marcus Rashford to the first team last season and is the embodiment of the United traditions of a commitment to youth and attacking football.
However, he was part of a static management team criticised by fans, pundits and former players for a dull style of play which ultimately cost Dutchman Van Gaal his job.
Led by executive vice-chairman Ed Woodward, United are now a different club to the trophy-winning juggernaut of the Sir Alex Ferguson era.
With a revised transfer policy of world-class marquee signings and the abrupt termination of Moyes and Van Gaal's contracts, along with the appointment of Mourinho, winning has been prioritised ahead of longevity and long-term planning.
Despite one senior member of the club's hierarchy saying last year that Giggs "triple-ticked" many essential components needed for the manager's job, he remains unproven at the highest level.
Giggs has made no secret of wanting to manage United in the future and it was thought the club had a long-term plan with the same aim.
He is unlikely to be short of offers from chairmen ready to give the Welshman his first permanent managerial role, having passed all the relevant Uefa coaching qualifications and served his apprenticeship at one of the biggest clubs in the world.
Wales manager Chris Coleman has previously said Giggs could benefit from the "spark" of stepping out of his "comfort zone" at United, backing him to "cut his teeth somewhere else and prove he has what it takes to go back there".
Similarly, United goalkeeping great and former team-mate Peter Schmeichel said Giggs should "learn the trade" of management at another club.
Ryan Giggs has ended his 29-year stay at Manchester United by agreeing a settlement with the Old Trafford club.