Fernandes has been linked with moves to Bayern Munich and Saudi Arabia and the United captain previously said he would consider his future after Euro 2024, which starts next month.
The 29-year-old ignited further talk of a United exit last week when he said: "If for some reason they don't want to have me, I will go, but if they want me I will stay."
With United manager Erik ten Hag fighting to avoid the sack after his side finished eighth in the Premier League -- their lowest final position since 1990 -- the club face an uncertain future.
Fernandes, who has two years left on his contract, wants to see signs of improvement on the pitch as he contemplates his long-term plans.
"I love to step out at Old Trafford more than anything in the world. I don't want to leave. This has always been my ultimate dream," he wrote in a letter to fans in The Players' Tribune ahead of United's FA Cup final clash against Manchester City on Saturday.
"I just want my expectations to fit with the expectations of the club. If you go talk to any fan, they will tell you the same thing.
"We want to compete for the league. We want to be playing Champions League football. We want to be in cup finals. That is the standard. That is what I want. That is what you all deserve.
"I just want to keep fighting. I want to be here."
Ten Hag's side have a chance to finish their turbulent campaign on a high by winning the FA Cup for the first time since 2016, at the expense of Premier League champions City.
Fernandes accepted responsibility for the team's poor season, in which they finished bottom of their Champions League group.
"I know that it has not been an easy season for you. We have not been up to the standard that you deserve," he said.
"There were so many moments when we won a massive game and it felt like 'OK, now we are going to build from this' and it just never happened.
"After this difficult season, it is my responsibility to give more. It starts with me and it starts tomorrow.
"We have to give everything in this last match against City, and we have to move forward."
