Stoke announced that reserve goalkeeper Jack Butland had put pen to paper on a new deal on Friday, and manager Hughes is keen for first-choice stopper Begovic to follow suit.
"Jack Butland is one of the top young keepers in the country and we are delighted that he is staying here with us," the Welshman explained.
"It's not his ambition to sit here and be an understudy - he wants to get in the team and he is continuing to get better.
"We are fortunate to have a top-level group of keepers. Asmir is one of the best in the league and at the moment is ahead of Jack.
"They all push each other though, Thomas Sorensen included, and as a player you need that. Competition is good in any area of the side.
"We will sit down and speak to Asmir in the summer. He wants to focus on the games first, and rightly so. We are comfortable with that.
"I know Asmir is happy here and is pleased with our progression. We hope they will be good discussions with him.
"We have re-contracted Jack and our intentions are to also re-contract Asmir too. We are trying to look after business in the correct way."
Begovic's current deal expires at the end of next season.
