Begovic, 27, is out of contract in 2016 and has been tipped to move on at the end of the 2014-15 Premier League campaign, with Tottenham, Liverpool, Real Madrid and Inter among the reported suitors.
Stoke manager Mark Hughes announced last week that the club will enter negotiations over a new contract in the close-season.
Wilson, who emerged through Portsmouth's youth system alongside Begovic, said Stoke must keep hold of the "terrific keeper".
"I've seen him since he was 16 and we came through the same youth team together," said the 27-year-old Republic of Ireland international.
"I even said back then that he was going to be a top-class keeper and he's proving it so far.
"The club would love to tie him down as soon as possible. It's crucial to keep your best players to make those extra strides forward.
"The manager and everyone in the dressing room would love to keep him for the foreseeable future."
Wilson added: "Asmir wants to play at the highest level and that's fair enough.
"As a player, that's ambition. But right now, he is only concentrating on Stoke. He's a very loyal guy and forever how long he is at the club, he will be 100 per cent focused."
