His continued good form despite the team's struggles this season has led to speculation about a move to Manchester United -- the club his father, Peter, captained to the 1999 Champions League title.
"Yes, I do (expect him to stay)," Shakespeare told a news conference on Thursday when asked about Schmeichel's future. "He's in the form of his life."
Schmeichel recently changed his agent, fuelling speculation in the British media that he might soon change clubs, but Shakespeare denied that was the case.
"There's been a lot of speculation this week, but he's totally entitled to change his agent," the manager added.
"But what I would say is he's been very professional, his performances have shown that. He has a very strong character, and I expect him to carry on in the same way.
"I don't feel the need to talk to him, because he's perfectly entitled to change his agent when that contract comes to an end."
Leicester, who were flirting with relegation earlier in the season, have recovered impressively since Shakespeare took over from Claudio Ranieri in February and are currently in 11th place on 40 points.
They host 13th-placed Watford in the league on Saturday and Shakespeare said his team would push hard for a top-half finish.
"I want to, the players want to and the club want to finish as high as possible in the league. That is the aim," he added. "I'm a big believer in momentum.
"We've got three home games left and the fans have been excellent this season. We want to finish on a high."
Skipper Wes Morgan (hamstring) is back in light training but not fit enough to face Watford.
(Reporting by Simon Jennings in Bengaluru)
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