Louis van Gaal conceded that Manchester United may need to win the Europa League to qualify for next season's Champions League after a 2-1 defeat at Sunderland on Saturday.
Bidding to close the six-point gap separating them from the Champions League places, United fell behind in the third minute when Wahbi Khazri's free-kick from wide on the Sunderland left crept in.
Anthony Martial levelled before half-time, but an unfortunate 82nd-minute own goal by United goalkeeper David de Gea gifted Sunderland the points and increased the pressure on embattled United manager Van Gaal.
"It will be very difficult now (to finish in the top four) because we give five points away, Chelsea and now," Van Gaal said, referencing last weekend's 1-1 draw at Chelsea.
United face Danish side FC Midtjylland in the Europa League last 32 next week and Van Gaal agreed when it was put to him that his side's best chance of reaching the Champions League is to win the competition.
"After this match, I think that is the best route," he told BT Sport at the Stadium of Light.
Compounding United's misery, Italian right-back Matteo Darmian sustained a dislocated left shoulder that Van Gaal said would keep him out for "several weeks".
Khazri had set up Jermain Defoe for a late equaliser in Sunderland's 2-2 draw at Liverpool and he made an early impression on his home debut with a free-kick that eluded everyone to nestle in the bottom-right corner.
United equalised in the 39th minute when Martial artfully dinked in his 10th goal of the season after Vito Mannone had parried Juan Mata's shot.
But second-bottom Sunderland regained the lead when the unmarked Lamine Kone met Khazri's right-wing corner with a powerful header that found the net after Martial's attempted clearance came back off De Gea's elbow.
United were last week reported to have approached Jose Mourinho about succeeding Van Gaal.
The Dutchman has described the story as a media invention and when asked if he felt like Saturday's result had put him under more pressure, he replied: "No."
Mourinho's former club Chelsea continue to prosper under interim manager Guus Hiddink and extended their unbeaten run to 12 matches with a breezy 5-1 home win over Newcastle United that elevated them to 12th.
- Payet inspires comeback -
Diego Costa, wearing a mask to protect a broken nose, Pedro Rodriguez and Willian all found the net inside the first 17 minutes, with Pedro adding a second goal before substitute Bertrand Traore completed the rout.
Andros Townsend replied late on for Steve McClaren's Newcastle, who slumped back into the relegation zone.
"We tried to press from the first minute to surprise them, and I think we did," said Hiddink, whose side visit Paris Saint-Germain in the Champions League last 16 on Tuesday.
Chelsea are now eight points shy of the European places, but victory came at the cost of an injury to departing captain John Terry, who went off in the 38th minute.
"Hopefully, it is only a small muscular problem," Hiddink told Sky Sports. "We will see what will be the outcome tomorrow."
With Manchester City hosting Tottenham Hotspur and leaders Leicester City visiting Arsenal on Sunday, defeat was doubly frustrating for Manchester United as they had a chance to close on the top four.
That they held onto fifth place was only because West Ham United fell just short of a stirring comeback after coming from 2-0 down to draw 2-2 at Norwich City.
Dimitri Payet celebrated his new West Ham contract by scoring and then teeing up Mark Noble for a thumping 76th-minute equaliser, after goals from Robbie Brady and Wes Hoolahan appeared to have put Norwich in control.
Southampton crept past West Ham into sixth place on goal difference after stretching their unbeaten run to six league games by winning 1-0 at Swansea City courtesy of Shane Long's 69th-minute header.
Watford climbed to eighth after captain Troy Deeney scored twice in a 2-1 win at Crystal Palace, who momentarily drew level through new signing Emmanuel Adebayor's header.
Stoke City are ninth after a 3-1 win at Bournemouth in which record £18.3 million ($26.4 million, 24.2 million euros) signing Giannelli Imbula opened the scoring with an instinctive ninth-minute volley from 25 yards.
Ibrahim Afellay and Joselu swelled Stoke's lead early in the second half before Matt Ritchie hit back.
Everton lost ground on their rivals for European football after a 1-0 defeat at home to West Bromwich Albion, for whom Salomon Rondon chested in a 14th-minute winner.