The Scot has not featured for the Stadium of Light outfit since coming off the bench in his side's 1-0 victory over Stoke City on January 29.
But, with the striker set to return to training on Wednesday, coach Poyet is hopeful he could be in contention for this weekend's visit to the Emirates Stadium.
"He will start tomorrow with us," the Uruguayan told the Sunderland Echo. "Then, depending on how he feels, we can either accelerate him for Arsenal or make sure he is right for the final (Sunderland face Manchester City in the League Cup final on March 2).
"He's had a programme with the expectation that he will be able to play at Arsenal.
"I don't know Achilles injuries, but for people I have known, they are always worried about feeling something, so we'll have to see how he is.
"Perfection would be for him to play against Arsenal. I would like to have him for then, even if it's on the bench.
"That would be a great situation for him to feel part of the group again and playing.
"If not, I would like to think he will be alright for the final."
Fletcher has scored just three goals in 22 appearances in all competitions for Sunderland this season.
