Shaw has been handed a first senior call-up for England, ahead of next Wednesday’s friendly against Denmark.
The match, which takes place at Wembley, is the last fixture before Hodgson names his provisional squad for Brazil.
One of the biggest questions is whether 33-year-old Cole, who has 106 caps for England, will make it onto the plane, or whether Shaw will dislodge him.
With Cole struggling to make it into the Chelsea side, Le Tissier believes Shaw, 18, is the frontrunner.
“I think if Cole doesn’t play regularly for Chelsea from now until the end of the season, Shaw has a great chance to make the World Cup team.
“I would definitely take Shaw instead of Cole, if the Chelsea defender isn’t playing for his club.”
Le Tissier added: “I think Shaw is ready for England. He looks mature for an 18-year-old from the performances I’ve seen.
“He looks like he will be England’s full-back for many years to come. It is important to get the young lads involved early. If you are of top quality, it shouldn’t matter how old you are.
“There was a fair bit of talk around the club, before he made his debut, about how good the boy was.
“You could see how he was going to have a successful career. He had the presence on the pitch, which you don’t often get from a kid of a young age, and he influenced matches. For a full-back, this is a remarkable trait.”