The 18-year-old completed his move to Old Trafford last week, which saw him become the most expensive teenager in world football.
Former United striker Cole has claimed that while he thinks Shaw has what it takes to be a success at the club, the amount that has been spent on him is far too much.
"It's a hell of a lot of money," he is quoted as saying by the Daily Star. "I don't think too many are going to spend that on a young kid.
"I know that he's English as well, its a big plus that he is English, but I think that's why so many managers go for foreign imports as they're cheaper.
"He's going to be a very good player, he keeps improving, he'll be a good player."
Shaw has made three appearances for England and played in his country's goalless draw against Costa Rica at the World Cup.