The 26-year-old Scotland international has missed the last three games after being told to rest a niggling Achilles problem.
However, he is due to return to training on Tuesday, and that could put him in contention for Saturday's Barclays Premier League trip to Arsenal and, beyond that, the Black Cats' big day against Manchester City at Wembley the following weekend.
Poyet told the Sunderland Echo: "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 all right for the final."
Fletcher's last appearance came as a substitute in the 1-0 league victory over Stoke at the Stadium of Light on January 29, while he last started in the Capital One Cup semi-final, second leg clash with Manchester United at Old Trafford a week earlier.
With Ignacio Scocco having arrived from Brazilian club Internacional and Connor Wickham back on Wearside after his loan spell at Sheffield Wednesday, Poyet acted on advice to withdraw the player from the firing line in an effort to cure the problem without resorting to surgery.
Only time will tell whether or not that has proved successful, but if so, Fletcher could yet have a role to play at Wembley.
Poyet said: "He will start tomorrow with us. Then, depending on how he feels, we can either accelerate him for Arsenal or make sure he is right for the final.
"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 will have to see how he is."