Khan, 32, returned to boxing last April after 23 months away from the sport and secured two wins in 2018, BBC Sport Reported.
The Briton had an offer to face domestic rival Kell Brook in a bout, which his team had said would prove more financially rewarding.
But he will instead be a heavy underdog against Crawford, 31, in a bout slated for April 20.
A news conference is due to take place early next week to outline a venue, either in New York or Las Vegas. Both are cities in which Khan has boxed before – this time, though, he will go up against an American widely regarded as one of the sport's best pound-for-pound fighters.
Omaha-born Crawford has held world titles at lightweight, super-lightweight and welterweight, and has already been installed as 1-10 favorite with some bookmakers.
More notably, he was an undisputed champion at super-lightweight in 2017 when he overcame Julius Indongo to hold all four world titles in the division.
He moved up to the 147lb welterweight class to take his current title from Australia's Jeff Horn in June, and his most recent stoppage win over Jose Benavidez Jr. in October took his record to 25 knockouts from 34 wins.
Khan has faced disapproval in some quarters for his September points win over Samuel Vargas, where he was knocked down before recovering to take his record to 33 wins and four defeats.
And Crawford, who boxes from the tricky southpaw stance, represents a huge step up as Khan bids to become a world champion for a second time having held titles at super-lightweight between 2009 and 2011.