Shaw is being considered by Roy Hodgson for a place in the squad to face Denmark, at Wembley, next month.
That will be the final England game before the provisional 30-man group is selected for Brazil.
If teenage left-back Shaw does complete a meteoric rise by making it to this summer’s tournament, Davis is confident that he will take it in his stride.
“Without doubt, I think he could cope with it,” said the midfielder.
“He is such a down-to-earth lad and one of the best things about him is that he doesn't realise how good he is.
“You don't want to heap too much pressure on young players, but everyone knows the quality he has got.
“I think he is more than capable of going and doing a job for England. It would be great for him to get that opportunity.
“But he is so level-headed that I wouldn't have any concerns with him dealing with it.”
Davis, the Northern Ireland captain, added: “For his age, it is unbelievable the talent he has got.
“Barring injuries and, obviously, staying level-headed like he is, he will go on and reach the top. There is no doubt about it.”