Morrison has made an impressive start to the campaign, earning his first England U21 caps and scoring twice in the recent 5-0 victory over Lithuania at Portman Road.
Cole, during his first spell with the Hammers, and Arsenal winger Theo Walcott both earned their full international debuts as teenagers.
And the 31-year-old is confident Morrison is developing well with the World Cup in mind, and England manager Roy Hodgson keen to promote youngsters.
"He's 20 now," said Cole. "That's young but I think Theo went at 17 and I was 19.
"He's got a chance. I think any English players will be thinking they have a chance. Certainly the young ones because Roy has come out and said he's going to go with youth over experience.
"His attitude is spot-on. He loves football, which I look for in a player the most. He loves his football. You can see the joy when he's playing.
"You can see him listening when you're talking to him. He's got a great attitude and that's why he is where he is."