Sky Sports understands Middlesbrough are lining up a move for Chelsea midfielder George Saville.
Wolves were among those to express an interest in Harry Maguire, but claim to have accepted defeat some time ago.
A United scouting delegation was reported to have been in Zenica on Friday, where they watched Begovic helped Bosnia to a 3-1 win over Greece.
"I'm not under more pressure because United is watching me, I'm glad that they do because that mean I'm doing something good. It gives me more motivation to do better and better," he told Sport.ba.
"I'm glad that they do because that means I'm doing something good. It gives me more motivation to do better and better.
"I'm pleased that the big clubs are interested in me, but as I said many times - I'm happy at Stoke City because they gave me a chance to play in Premier League.
"We will see what will happen in the summer. In football you can never know what will happen."