The-25-year-old was linked with a move to the Red Devils during the January transfer window, while club scouts are said to have run the rule over him during Bosnia's 3-1 win against Greece on Friday.
"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 mean 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 the gave me a chance do play in Premier League.
"We will see what will happen in the summer. In football you can never know what will happen."
The arrival of Jack Butland at Stoke in the summer is expected to pre-empt Begovic's departure, something he admitted was likely to happen back in February.
The Red Devils were said to be keen on bringing in Begovic as a replacement for the inconsistent David de Gea, although since the close of the January transfer window the young Spaniard has emerged as a more reliable performer.
Arsenal and Chelsea have also been credited with an interest in the former Portsmouth shotstopper.
The Gunners are expected to be in the market for a goalkeeper in the summer, and could move to take Begovic to the Emirates Stadium to challenge the under-fire Wojciech Szczesny for a starting spot.
Chelsea meanwhile are in need of a number two to challenge Petr Cech at Stamford Bridge with Thibault Courtois seemingly staying put at Atletico Madrid for next season.
Previous reports have suggested Stoke may hold out for £15 million for their first-choice goalkeeper.