Stoke City boss Mark Hughes refused to allow Jack Butland to leave on-loan last week.
The People says England international Butland was keen to finish his campaign by adding to the experience he gained with Derby County and Leeds United earlier this season.
But Butland's potential move on an emergency loan before the window closed was ruled out as Begovic looks increasingly likely to be on his way this summer.
The Bosnia & Herzegovina international has just 12 months left on his existing deal and the Potters have moved this week to tie Butland to a new four-year agreement.