The Bosnian stopper has been one of the Premier League’s most consistent glovesmen over the course of the past few years, attracting attention from some of the division’s big hitters.
Both Manchester City and Chelsea are believed to have made approaches for his signature in recent times, but their efforts have been fruitless.
However, Real coach Carlo Ancelotti is hopeful that he can push through a summer deal.
The Italian is said to have made moves for the Potters star while in charge at Stamford Bridge, and has maintained an interest in him ever since.
With Iker Casillas no longer a key man at the Bernabeu and Diego Lopez edging into his the twilight years of his career, Begovic is being considered to become Los Blancos’ new number one.
The solid ‘keeper is valued at around £15m, and a good performance at the World Cup could persuade Real’s hierarchy to sanction a serious offer.
The 26-year-old is unlikely to be back on Chelsea’s radar this summer, with the London club set to make a major decision regarding the futures of Thibaut Courtois and Petr Cech, while City appear to have made Joe Hart their main man once again after an early season blip.