The 25-year-old made his first England appearance against Sweden in November after impressing for the Potters in the Premier League this season.
Shawcross, who made two appearances in a six-year spell at the Red Devils, has been linked with a move back to Old Trafford, but Adam is confident the Stoke skipper will sign a new long-term deal to keep him at the Britannia Stadium.
“When you’ve got quality like that, you need to keep it in your group and the manager has always stressed he’s one of our best players,” Adam told the Sentinel.
“This is a good club going forward and Ryan wants to be part of that. He’s been here for a number of years.
“He’s seen it from what it was to what it is now and he knows it’s progressing. It’s good to have people like that at the club.”
Adam believes Shawcross’ England call-up was long overdue, and the former Liverpool midfielder is delighted with the experience in Stoke’s back four.
“The call-up to the England squad is thoroughly deserved, probably 18 months late, but he has been excellent for this club,” he added.
“We’ve got an experienced back four. We’ve got Robert Huth, who’s played a number of times for Germany, Geoff Cameron, who’s played for America and Asmir Begovic, who plays for Bosnia.
“From the front to the back, we work as hard as anybody in this league and we try and get the rewards.”
Get enhanced odds of 18/1 on Andy Murray to win Wimbledon or 10/1 on Novak Djokovic when you open a Paddy Power account.