The managerless side will look to make amends for missing out on European football this term and their Grimsby humiliation in the Carabao Cup.
While they will be desperate to return to Europe via their league placing, they cannot win the the title, increasing the importance of a successful FA Cup run.
Interim United boss Fletcher said: “It’s an important competition and Manchester United is about winning trophies.
“I think we’ve had good success in the competition in recent years – won it a couple years ago, lost in a final.
“It’s an amazing competition. It’s the first trophy that I won as a player. It’s a special tournament.
“With the season so far – not in Europe, out of the League Cup, Premier League obviously – the FA Cup’s a trophy that we should be vying to win and giving ourselves every opportunity to win.”
United enter the FA Cup sat seventh in the Premier League and within three points of the final guaranteed Champions League spot, as the club considers who to lead them through the remainder of the season.
Ole Gunnar Solskjaer and Michael Carrick are the frontrunners for the post, but Fletcher and Ruud van Nistelrooy are also in the frame.
Solskjaer was set for face-to-face talks with club chiefs on Saturday while Carrick met with the United hierarchy earlier in the week.
On the pitch, Bruno Fernandes, Harry Maguire and Kobbie Mainoo returned to the squad for Wednesday’s 2-2 draw at Burnley, with the latter coming off the bench after recovering from a calf complaint.
The 20-year-old has been frustrated this season by a lack of game-time, with Amorim failing to hand him a single Premier League start in a campaign that began with his request to leave on loan being blocked.
“He seems in a good place – Kobbie doesn’t give you much, so you wouldn’t know if he was in a good place or not so good,” Fletcher said with a laugh. “He’s like that. That’s his natural way.
“But I know him well and I’ve known him for a long time. I’ve seen him around throughout the season and I’ve had conversations with him in general, but he’s in a good place."
“I know Kobbie and I think he knows me and he knows (interim first-team coach) Travis Binnion, so he’s comfortable with his environment, he’s in a good place, he’s trained well.”