Kazakhstan's Golovkin is unbeaten in 35 professional fights with 32 knockouts.
British middleweight champion Chris Eubank Jr had called out Golovkin, but Brook has got in before him.
"When this fight was offered I accepted in a minute. I'm a huge welterweight and I will carry my speed, accuracy and power through the weights," said Brook.
"This is the ultimate fight, one of the biggest international fights this country has seen in many years.
"You have seen some huge names run from Golovkin but I'm running to him. I'm a special fighter and I will show the world on 10 September."
Sheffield's Brook, 30, is following in the footsteps of compatriot Amir Khan, who also jumped two weights  to challenge Mexico's then WBC middleweight champion Saul Alvarez in May.
That fight was at a catchweight of 155lb, five pounds below the middleweight limit, but it is yet to be confirmed if Golovkin will make any concessions.
"I'm very excited to be fighting in front of the great British boxing fans and promise another 'big drama' show," said 34-year-old Golovkin, the WBC, WBA and IBF middleweight champion.
Golovkin has never boxed in Britain before but has beaten two British fighters in world title defences - Matt Macklin was knocked out in three rounds in 2013 and Martin Murray lasted 11 rounds in 2015.
Both Brook and Golovkin have been hunting huge-money fights in recent years but have been repeatedly frustrated.
Brook, who is unbeaten in 36 pro fights with 25 knockouts, won his IBF title by outpointing Shawn Porter in 2014 but what seemed a natural fight against Khan failed to materialise.
Golovkin won his first world title in 2010 but has struggled to find willing opponents, with Alvarez preferring to give up his WBC belt in May rather than face him.
"I cannot tell you how excited I am for this fight," said Brook's promoter Eddie Hearn.
"You have the number one welterweight in the world against the number one middleweight in the world - 36-0 v 35-0, two of the best on the planet. It's what big-time boxing is all about."
After defending his British belt against Tom Doran last month, Eubank Jr said he was keen on a match with Golovkin.
It remains to be seen why that fight fell through. Golovkin recently suggested Eubank Jr was presented with a contract and failed to sign it, but Eubank Jr denied that.
Eubank Jr said on Twitter: "I haven't been shown one contract since my fight with GGG was announced. I'm still ready & waiting to sign. Fighting Brook is a joke."
Britain's IBF welterweight champion Kell Brook will challenge undisputed world middleweight champion Gennady Golovkin in London on 10 September.