The Bosnian international has been heavily linked with the Red Devils for some time, but is now believed that their local rivals are prepared to take them on in the race for his signature.
Begovic has enjoyed a fine season so far with the Potters, helping them establish one of the finest defensive records in the top flight.
This has alerted United, who are believed to be looking to replace current number one David de Gea.
Sir Alex Ferguson has reportedly lost patience with the young Spaniard, who has made multiple errors since his arrival from Atletico Madrid.
He was billed as the man to fill the void left by Edwin can der Sar, but has failed to match the performances of the Dutchman.
Begovic is seen as the ideal replacement by Fergie due, in part, to his eligibility to play in this season’s Champions League.
City are also interested, and see the ex-Portsmouth stopper as the perfect man to keep Joe Hart on his toes.
The England ‘keeper has been short of his best this term, leading to speculation that he has become too comfortable at the club.
His boss Roberto Mancini is aware that his side must improve to challenge United for the title and has highlighted Begovic as a potential addition.