With seven games still to play in the Premier League, United are currently fifth in the table - a position which would see them miss out on a place in next season’s elite European club competition.
The Red Devils, who face title-challengers Tottenham at White Hart Lane this weekend, sit one place and one point behind neighbours Manchester City in a battle for the final qualifying place.
Right-back Darmian joined the club last summer from Italian outfit Torino for a fee believed to be around £12.7 million.
The 26-year-old has gone on to make 25 league appearances under manager Louis van Gaal and is hoping the Red Devils will manage to end their campaign with a cup success.
United were eliminated from the Champions League at the group stage before exiting the Europa League to Liverpool in the last-16.
However, the club remain in with a chance of silverware this term, as they prepare to face high-flying West Ham in the final four of the FA Cup next Wednesday.
Darmian admits expectations are high at Old Trafford and the players are trying their best to achieve a vital top-four finish.
"It's normal that when you play for Manchester United there is a lot of pressure, but we know that and we try to do our best,” he told Sky Sports.
"I think Manchester United is the biggest club in the world so there is a lot of pressure. They expect that we win all the games.
"Now we want to be in the top four at the end of the season. We have a big chance to win the FA Cup, so we want to improve and work, and if possible win a trophy.
He added: ”Yes of course [top four is minimum requirement]. We want to be in the top four at the end of the season and the club works for this.”