Fletcher scored Middlesbrough’s fifth goal as they cruised into the fourth round but remains a peripheral figure under manager Pulis, with his only Championship start having come at the start of last month.
Hull are keen to take him to the KCOM Stadium this month, and Pulis revealed that a number of other Championship clubs have also made formal inquiries.
Middlesbrough have already lost Martin Braithwaite this month, with the Denmark international having joined Spanish side Leganes on loan, but Fletcher could also be sacrificed as part of a major overhaul.
Pulis said: “What’s his situation? He’s just played a game for us and, hopefully, that will give him the confidence to move on and improve.
“There’s not just Hull that are interested, there are one or two other clubs as well. Will he stay here? We’ll see what happens.
“It was great that he got his goal, but I thought he played well in the first half, too. There were one or two who didn’t do as well in the first half, and got better in the second half, but I thought Fletch was one of our better players in the first half as well. I’m really pleased with him.”
Middlesbrough struggled in the first half, but a tactical tweak at the interval had a major impact, with Lewis Wing replacing debutant Rajiv van La Parra to immediate effect.
Pulis said: “The application of the players was very good. In the first half, we played with five forwards really with Tav (Marcus Tavernier) and Raj (van La Parra) out wide. They all pushed on, and if anything, I’ve made a mistake doing that because it blocked all the spaces we were really trying to use.
“Bringing Wingy on, and then pulling him and Grant (Leadbitter) out to play a bit more, sucked them on to us, and then there more spaces for us to play in.
“I think the big thing is we’ve scored five goals, and I’m really pleased with the attitude and application but we were playing against a League One side, not against a top Championship side.”
Middlesbrough: Konstantopoulos, McNair, Fry, Ayala, Friend, Fletcher, Leadbitter, Tavernier (Chapman 75), van La Parra (Wing 46), Gestede (Walker 74), Assombalonga. Unused substitutes: Clayton, Wood-Gordon, Lonergan, Spence.
Peterborough: O’Malley, Naismith, Tafazolli, White, Lafferty, O’Hara (Godden 58), Woodyard, Reed (Cooke 75), Ward, Toney, Dembele (Cummings 80). Unused substitutes: Chapman, Bennett, Cooper, Denton.
Referee: J Linington (Isle of Wight).