The Bosnian stopper has been on United's radar this month but a move in the summer could be more realistic as other top clubs are also interested to sign the 25-year old.
United boss Sir Alex Ferguson is not completely convinced with his current goalkeepers David de Gea and Anders Lindegaard. De Gea, 22, is yet to settle in England and has been linked with a move to Real Madrid or Barcelona.
Manchester City have also shown keen interest in Begovic, with a view to give tough competition to Joe Hart.
Meanwhile, United have been eyeing Liverpool keeper Pepe Reina but face heavy competition from Barcelona. United are close to signing 19-year old Dean Santangelo from Augsburg but not as a direct replacement for De Gea.
Begovic's choice to join United is clear as he doesn't intend to face direct competition from Hart at City. At Old Trafford, the Potters No. 1 will directly replace De Gea in goal.
United boss Ferguson has looked at other options across Europe with Fernando Muslera from Galatasaray, Newcastle United's Tim Krul and Sunderland's Simon Mignolet.
Stoke manager Tony Pulis will demand nearly £15 million for Begovic, who is under contract with the Potters until 2016.