The Bosnian stopper has been linked with United since the January transfer window, and Sir Alex Ferguson was believed to be considering a move to bring in competition for David De Gea.
The Spaniard was struggling as United number one at that stage of the season, but ever since his performance at the Bernabeu against Real Madrid in February, De Gea has been superb.
And the report claims United's interest in Begovic has now cooled, to such an extent that Fulham are now in pole position to sign the Stoke City keeper this summer.
Begovic is thought to have become unsettled since Tony Pulis agreed a deal to bring Jack Butland to the club next season. The England international remains on loan at Birmingham City until next term, but will challenge Begovic for the number one spot when he joins.
And Fulham are looking for a long-term replacement for Mark Schwarzer. The veteran keeper has been offered a one-year extension at Craven Cottage, but wants assurances he'll be first-choice next season.
But Martin Jol has prioritised the summer signing of a keeper, and the £4m-rated Begovic is viewed as the best value-for-money target.
As well as Manchester United, Arsenal have also been linked with a move for the Bosnian. Wojciech Szczesny has dropped to the bench in recent weeks as Arsene Wenger has insisted he's mentally tired, and Lukasz Fabianski has taken his place between the sticks.
But the decision to drop Szczesny has fuelled speculation Arsenal could be in the market for a keeper this summer, with moves for Begovic and Barcelona's wantaway keeper Victor Valdes under consideration.