The Scotsman has maintained that he will not discuss his future at the club until the end of the season, which Stuart believes is a sign that Moyes wants to try a new "challenge" elsewhere.
"My head says that this summer he might think the time is right to take on another challenge," Stuart told the Liverpool Echo. "My heart says we want him to stay because he's been a huge part of a successful period of time in terms of Premier League finishes for the club.
"We just haven't quite managed to turn it into a trophy which is the one aspect of disappointment I'm sure David shares with everyone connected to the club.
"The fact that the contract hasn't been signed, and he's been here such an amazing amount of time does make me wonder whether he wants a new challenge."
Moyes is in his 11th year with the Merseyside outfit.