Abbott feels it is too early for him to move to the Premier League - and will urge him to continue his footballing education in Cumbria.
"I'm sure Newcastle aren't the only club looking at him. They need to be on the ball and if big clubs are doing their job they will be looking at him," he told the Sunday Sun. "I'll be surprised if there aren't a lot of clubs looking at our lads at the moment.
"As for Gary, I don't think it is the right time for him to go to a club like Newcastle. We love him here, that's why we got him on a long-term contract and I think he is improving so much week by week.
"My advice to him has always been 'Get 60 starts under your belt, score your goals and learn your trade with us'. I think he's sensible enough to see that.
"But I know that when a club like Newcastle looks at you, you're going to be flattered. If they actually came to us with an offer, it would be something we'd look at like we always do.
"But Gary is a terrific lad and he's sensible enough to know where he needs to improve. We believe he has a great future with us."