Begovic has established himself as one of the Premier League's most consistent keepers since joining from Portsmouth three years ago and is closing on 100 appearances for Stoke.
He has been mooted as a replacement for David de Gea at Old Trafford and there have also been claims that Manchester City want to sign the 25-year-old as a back-up for Joe Hart.
The Bosnia international insists he has not made a decision on his future, but believes Stoke's recent signing of Jack Butland will lead to his departure from the Britannia Stadium at the end of the season.
"I read in the papers that I said no to City and yes to United," Begovic told Oslobodjenje. "But the truth is, I haven't decided on anything.
"I didn't say yes or no to anybody. I haven't signed anything - no contract, no pre-contract. But I believe that Stoke didn't buy Butland to be a reserve so I think I'm going to be sold in the summer.
"To whom? I can't move to a smaller club and I also know that some top clubs are interested in me. They are also the only ones who can afford the fee.
"Now I'm just happy that January is over. It was a hard month for us, so many rumours. Now, we can focus on our games again."
England keeper Butland will link up with Stoke in the summer after the teenager was loaned back to Birmingham for the rest of the season last week.