30-year-old Belgian global Fellaini has been at Old Trafford since making the move from Premier League rivals Everton in 2013.
Mourinho has had nothing but praise for Fellaini since the pair began working together and he very much hopes an agreement can be reached.
"And I respect the board, because the decisions and discussions are under their control, so I just wait, wishing they will have an understanding".
Rumour has it that the Belgian wants more than the £120,000-per-week wages United are reportedly offering - at the age of 29 this is likely to be the last "big" contract he can command - while there may also be some discussion over the length of any deal.
The 30-year-old has endured a hard relationship with United supporters at times but has become an influential player under the Portuguese.
Fellaini has responded with improved showings over the first few months of the current campaign, and has already racked up 4 goals in 10 appearances between the Premier League and Champions League, just 4 of which came as starts.
"He's an important player for me - a lot more important than you can imagine", he said back in September.
Mourinho has admitted that another fans' favourite, Zlatan Ibrahimovic, is still some way from being able to start a game after his dramatic return from career-threatening knee surgery.
"If the team is winning, you need him on the pitch for the last 15, 20 minutes because his qualities are adapted to both situations".
"We have four competitions to play and let's go step by step, it's the only thing we can do", added Mourinho, whose side enter the weekend eight points to leaders Manchester City.
'It's an interesting week because I think it's the first time we have three Premier League matches in one week, with a couple of days in between matches.