The three-time European champions sneaked past the Danish side 1-0 at Old Trafford last month thanks to a last-gasp penalty save from goalkeeper Andre Onana.
Heavy defeats by Manchester City and Newcastle have since ramped up the pressure on manager Erik ten Hag, which was only eased slightly by a 1-0 win at Fulham on Saturday.
United are third in Group A, six points behind leaders Bayern Munich and one adrift of Galatasaray.
If they fail to win in Copenhagen, Galatasaray could be in a position knock United out when the sides meet in Istanbul on Nov. 29.
Recruitment
United fans have frequently voiced intense frustration at the running of the club by the Glazer family, but Ten Hag has still enjoyed a huge transfer budget.
More than £400 million ($497 million) has been spent in the past three transfer windows, with few obvious successes.
Former Ajax winger Antony, bought last year for £86 million, has rarely lived up to his huge price tag, scoring just eight goals in 55 appearances.
Mason Mount, like Antony, has regularly found himself on the bench in recent weeks just months after a £55 million move from Chelsea, and has yet to score for his new club.
Onana's penalty save against Copenhagen was atonement for costly errors against Bayern and Galatasaray as United lost their opening two group games of a Champions League campaign for the first time.
One of the criticisms aimed at Ten Hag is that he has been too reliant on the Dutch market, which he knows well.
Eight of his signings have previous experience of playing in the Eredivisie but many of them have struggled with the extra demands of playing for one of the world's biggest clubs in the Premier League.
Injuries
Ten Hag is frequently criticized for failing to impose an identity and style of play, nearly 18 months into his reign, but he has been hampered by injuries to key players.
United performed impressively in the Dutchman's first campaign in charge, ending a six-year trophy drought and finishing third in the Premier League.
This season, long-term absences for Luke Shaw, Tyrell Malacia and Lisandro Martinez, as well as injuries to Raphael Varane and Aaron Wan-Bissaka, have stretched his defensive resources to breaking point.
In a further headache for the former Ajax manager, midfielder Casemiro has been ruled out for "several weeks" with a hamstring injury.
Veteran center-back Jonny Evans, brought back to the club as defensive cover, believes the absence of so many players has destabilised the squad.
"I think the manager's found out difficulties having to sort of work on the team, on tactics and having to change the personnel quite a lot with only two days in between games," he said.
Misfiring stars
Manchester United's marquee players have failed to deliver this season, with Marcus Rashford, Casemiro and Bruno Fernandes all well below par.
Rashford's decline is the most stark -- he has just a solitary goal in 14 matches, compared with 30 for the club last season, when he found a new level.
Ten Hag says the England forward is "very motivated to put things right" but he desperately needs his main man to find form quickly.
Brazilian international midfielder Casemiro, 31, has failed to match the high standards he set during an impressive first season at United.
Casemiro is top scorer, with four goals in all competitions, but he has struggled with the defensive side of his game, leaving a leaky defense even more vulnerable.
Portugal midfielder Fernandes has often cut a frustrated figure and has not been a consistent creative force for a side that have managed just 12 goals in 11 Premier League games this season.
Ten Hag will hope his late winner against Fulham on Saturday sparks a revival for the player he made captain in July.