Bairstow, 22, has been named in the England squad that is due to face the West Indies later this month for the first time.
He has been tipped to force his way into the starting lineup against the Windies and Bresnan believes that his Yorkshire teammate could go on to hold down a position in the team for years to come.
"He's good at everything," Bresnan told the London Evening Standard. "He plays golf well, he played hockey and rugby union to a high level when he was younger, so he is a brilliant talent and definitely one to watch. I'm a bit jealous to be honest.
"He seems to relish the big occasion — it brings out the best in him — and there is none bigger than your Test debut.
"He is not really fussed about things and is just quite a confident, well-rounded young bloke. If he does score runs, there's no reason why he shouldn't stay in there."
Bairstow has featured for England at one-day international and Twenty20 level.