The Bosnian has been out of the side for the last couple of matches, with young English goalkeeper Jack Butland doing well in his absence.
The former Portsmouth stopper has been an excellent signing since arriving at the Britannia in 2010,and has made 171 appearances for the Potters in that time, as well as scoring a goal against Southampton last season. This form has seen his name linked with both Manchester United and Tottenham in recent months, according to the Mirror, two clubs whose own goalkeepers are also subject to transfer speculation.
The situation is precarious for Stoke because Begovic's deal expires next summer.
But in quotes on Sky Sports from his conference ahead of a trip to Burnley on Saturday, Hughes made it clear that the club are attempting to tie the 27-year-old down to a new long-term contract.
"It's something that will be resolved, hopefully to a positive conclusion for ourselves," said the former Wales boss.
"I want Asmir to stay here. We feel we can offer him a good platform. We’re a good option for a lot of players now including Asmir, who’s been outstanding in his time here and we want that to continue
"We’re conscious that he’s in the last year of his contract, and we’ll discuss that situation with Asmir and his people.
"The plans is to sit down with Asmir and his people at the end of the season and we’ll have an honest discussion and see where that takes us."