Schmeichel Jr signed terms to 2023 last week.
"There's been a lot of interest in Kasper obviously but he feels really at home at Leicester," said Peter.
"To move away from that would have had be somewhere very special. There are not a lot of spaces at the eight top clubs in Europe and you certainly don't want to go somewhere as a number two.
"I'm pleased he's committed his future to Leicester. They're a good fit. Besides, he's not someone who'se in it for the money.
"For him, it's all about playing. He's gone the hard route and that's what has carried him to where he is now. He's in a really good place. At the World Cup, he didn't go out to show the world how good he was. That's when it can go wrong. He went out to do his job."