Belgium striker Lukaku suffered an ankle injury during last month's 2-1 win over Arsenal and has sat out United's past two Premier League matches.
Mourinho's men failed to score on either of their trips to Brighton and Hove Albion and West Ham, underlining the 24-year-old's importance to the cause following 27 goals across all competitions in his first season at Old Trafford.
"We hope he can play in the final,” Mourinho told reporters, with Lukaku certain to miss Sunday's Premier League curtain call versus Watford.
"He is in Belgium, having his treatment with communication between the doctors in Belgium and our medical department.
"We are just waiting to see if it's possible for him to be involved in the final. If not starting, then at least on the bench."
Michael Carrick will start as captain against Watford in his final appearance for United before joining Mourinho's backroom team for next season.
"Carrick had already been decided," the Portuguese added. "He's going to start the match as a captain and he's not going to end the match because I want him to leave the pitch alone and I want him to feel what the United supporters feel in relation to him.
"I am not so sad as I normally am when my players end their career because he is going to be with us. He is going to be even closer to me than he was as a player so it is just an emotional moment because it is the end of a career of a fantastic player."