However, according to The Sun, the Potters are requesting an £8 million fee plus a player on loan for their impressive shot-stopper, who would act as number two behind goalkeeper Thibaut Courtois should he complete a move to Stamford Bridge.
21-year-old striker Bamford won the Championship Player of the Year award last season after scoring 19 goals for Middlesbrough while out on loan.
After such a successful season, he is unlikely to accept a fringe role at Chelsea this time round, meaning Stoke could be an attractive destination for him.
Meanwhile, Loftus-Cheek is tipped to have a big future in west London, with Mourinho suggesting he will feature in the first-team this season.
However, he faced heavy criticism from the Portuguese head coach after a lazy performance during the club's post-season tour in Sydney - Mourinho may now feel he needs more senior experience before featuring regularly for the Premier League champions.
The third of the trio, Moses, is considerably more familiar to Potters fans.
Moses enjoyed a stint at the Britannia Stadium last season, where he scored four goals in 23 appearances, and appears highly unlikely to remain at Chelsea past the summer.
Nonetheless, the reported arrival of Xherdan Shaqiri, as per the Sentinel, could end the need for the Nigerian. If anything, the club could probably agree a separate deal for his services.
Who is most likely to arrive at the Britannia?
If the Potters are happy to part with the services of star man Begovic, you would expect Mark Hughes to take full advantage of the situation by luring a player who would otherwise be unavailable.
Moses looks set to leave Stamford Bridge regardless of any deal for the Stoke goalkeeper, so it seems wise for the Welshman to target either Bamford or Loftus-Cheek.
Despite his pre-season displays, Loftus-Cheek is likely to play a part for Chelsea next term, while Bamford will be left seeking first-team football elsewhere; something Stoke are able to offer.