The 25-year-old is regarded as one of the brightest young goalkeepers in the Premier League, after a number of impressive performances for the Potters over the past couple of years.
Begovic was linked with a move away from the Britannia Stadium in the January transfer window, with Stoke manager Tony Pulis already planning for life after the Bosnian international, following the successful capture of Jack Butland.
The England youngster was immediately loaned back to Birmingham City for the remainder of the 2012/13 campaign, but is expected to take over as Stoke number one if, as expected, Begovic departs this summer.
His sale could break the British transfer record for a goalkeeper, with Stoke confident of pushing Begovic's asking price closer towards £20million in the event of a three-way auction with Manchester United, Chelsea and Arsenal.
Manchester United shattered the previous record when they paid £18.3million to bring David De Gea to Old Trafford from Atletico Madrid in 2011, but Sir Alex Ferguson is considering bringing in another experienced goalkeeper to provide competition for the Spaniard.
Anders Lindegaard has played a bit-part role in the Red Devils' first-team and is expected to leave at the end of the season.
Begovic is ambitious and has set his sights on playing Champions League football, but at the same time is eager for regular action, so Manchester United might not be the best move for goalkeeper.
Arsenal could be a more attractive option with Arsene Wenger reportedly on the lookout for a replacement for current first-choice stopper, Wojciech Szczesny. The Pole has endured an indifferent season at the Emirates Stadium, and lost his place in the Gunners first-team to the returning Lukas Fabianski for last week's Champions League victory over Bayern Munich.
The Bundesliga giants - who booked their place in the quarter finals of Europe's top tier competition at the expense of Arsenal - have also been linked with Begovic. However, he would face a serious fight to dislodge long-term custodian Manuel Neuer at the Allianz Arena.
At Stamford Bridge, he would be presented with the same situation, as Petr Cech continues to keep goal for Chelsea, while on-loan star Thibaut Courtois looks like a ready-made replacement for the Blues, as he continues to excel in Spain, with Atletico Madrid.
Chelsea actually tried to sign Begovic in 2010 but had a £4million bid rejected by Stoke. They are still in a strong position, and don't need to sell, but the general feeling is that the club would not stand in the former Portsmouth star's way if he wants to leave.