Fletcher scored Middlesbrough’s fifth and final goal as they cruised into the fourth round of the FA Cup, but remains a peripheral figure under Pulis, with his one and 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 of the players. We played really well, especially second half. But we’re playing against a League One side, we’re not playing against a top Championship side.”
Peterborough boss Steve Evans felt his side did not deserve to finish on the end of a five-goal hiding, with Ivan Toney having missed a glorious first-half chance when the game was still goalless.
Evans said: “I’ll focus on the first 45, we gave them real problems and should have been two or three ahead at half-time. We missed some chances we’d have expected to take.
“You realise Tony is going to have a reaction in the dressing room, they changed the system but we have to defend properly.
“You’ve seen what’s happened out there today. When they get the second and then the third it becomes a long day against such a talented squad.
“It’s the best we’ve played in 10 months under me (in the first half). I thought we were terrific, made good chances, nullified their threat but you have to be capable of taking chances against any side in the Championship.”