The Bosnia international has regularly been linked with other clubs during his time at the Britannia Stadium, and although his current deal has just over 12 months to run, Scholes believes they can match the 27-year-old's ambitions.
"Clearly Asmir is a terrific goalkeeper and we are very keen to keep him at the club," he told The Sentinel.
"I think when you've got extremely good players it is only natural that there will be speculation about their future.
"These players are also very ambitious and I include Asmir in that
He is a very focused, very talented and very ambitious young man.
"He wants to make sure he can play at the very best levels his abilities will enable him to and to have a very good and successful career in the game .
"We hope that we can fulfil all of those things at Stoke City."
Begovic's deputy Jack Butland recently extended his contract until 2019 while manager Mark Hughes also agreed new terms on an extension.
Scholes said that, and the recruitment which has taken place since they opted to change their manager for the start of last season, highlighted how they were planning for the future.
"If you look at the deals we have done since Mark Hughes arrived then almost without exception they have been good deals and very good footballers," he said.
"It's about making sure you are adding to the quality and improving all of the time, but we have also got a strong eye on making sure that we have got a depth of squad.
"We have to make sure that when we have injuries to the incredible scale we have had this year that the players are able to come in and do a good job - and the players who have come in this season have done an exceptional job.
"You also have to make sure that you are planning for next year and the years beyond.
"Signing Jack Butland to an extended contract is a statement that we are in the Premier League for the long term."
Source : PA
Source: PA