Cook spent three months on loan at Carlisle last season, scoring four goals in 14 appearances, and Abbott has made no secret of his desire to bring the 22-year-old to Brunton Park on a permanent basis.
"Everyone knows he was here and we think he did well and we also know he enjoyed his spell with us," Abbott told Carlisle's official website. "He has a lot on his plate and a lot of things to digest, the biggest of which is the fact that he's just been let go by Sunderland.
"He has to get over that disappointment and it's not great etiquette as a manager to be banging his door down the day after he gets that kind of news.
"He's a local boy and Sunderland are his club so he'll be disappointed with that. What we've done so far is make sure his people know, in a delicate way, we are interested in possibly doing something and we can take it from there.
"I don't think it would do anybody any good to go steaming in at this moment in time."