Publicly, United continued to back Mourinho on Tuesday and there is little chance of him walking away from a £15million-a-year contract.
Woodward does not want to make a third change in less than four-and-a-half years but will come under enormous pressure if United suffer a third straight defeat on Sunday.
Mourinho continued to show the strain after Monday's 3-0 loss to Spurs at Old Trafford when he walked out of his post-match press conference after demanding respect for having won three titles with Chelsea.
Mourinho has not lost the dressing room but morale has suffered in recent months and it was noted that only eight players were on the team coach from Carrington to the Lowry Hotel for the Spurs game.
Others made their own way to the hotel, with Paul Pogba arriving in a chauffeur-driven Rolls-Royce.
United's worst start to a Premier League season in 26 years follows a turbulent summer in which Mourinho has been at odds with Woodward and the board over signings.
Insiders say there was an air of resignation among Mourinho and his backroom staff following the shock 3-2 loss to Brighton.
He remained calm with his players after they gave a more encouraging performance against Spurs but conceded three goals again as the 55-year-old suffered the heaviest home defeat of his career.
Woodward was said to have been in an upbeat mood when he came into the dressing room afterwards and spoke with Mourinho at some length, but his No 2 Richard Arnold looked suitably despondent.
Mourinho is understood to have welcomed the outspoken comments from TV pundits and former United favourites Gary Neville and Paul Scholes last week, placing some of the blame on Woodward and the erratic Pogba.
Unhappy fans have paid for a plane to fly a banner over Turf Moor on Sunday calling for the executive vice-chairman's head, but the reality is that Mourinho will be the one under threat if United lose again.
He tried to talk up the effort of the squad, who were given Tuesday off, and support of the fans on Monday as evidence of a united club.
However, strained relationships with a number of stars including Pogba, Anthony Martial, Luke Shaw and Eric Bailly over the past 18 months have left him dangerously short of support. Other players are not so openly hostile but would be happy to see him go.
In an attempt to show solidarity on Tuesday night, United goalkeeper David de Gea tweeted a picture of the squad linking arms prior to a match with the tagline: 'United now more than ever'.
Although Mourinho was unhappy at the lack of signings this summer, his inability to get the best out of the players already at the club has caused concern among the United hierarchy.
Pogba continues to underachieve, Alexis Sanchez has yet to meet expectations and £75m striker Romelu Lukaku has struggled after putting on too much muscle during the summer.
Defence remains the main issue, though, after Mourinho dropped Bailly and Victor Lindelof against Tottenham and gambled by playing midfielder Ander Herrera in a back three.
Mourinho's backroom team is no longer as harmonious as it once was, with the departure in May of his long-time assistant Rui Faria said to be a key factor.
