Jurgen Klopp's side thrashed the Red Devils 5-0 at Old Trafford, and there has been immense pressure on Solskjaer to get the sack in the wake of the defeat.
Reports this week have claimed that the Norwegian has three games to turn the team's fortunes around and keep his job. However, according to ESPN, a number of text messages sent by United players to Woodward were in support of Solskjaer.
The report adds that some of the texts included acknowledgements of the squad's failings, both collectively and at an individual level, but insisted that the manager still retains the respect of the dressing room.
Antonio Conte, Brendan Rodgers, and Zinedine Zidane have been linked to the hot seat. However, both Rodgers and Zidane have reportedly turned down interest, leaving only Conte in the spotlight for now.
Speaking ahead of the weekend's game, Solskjaer admitted his team are considering a tactical change to curb their abysmal run of form. "Of course, you think about it all the time, you think about tactical tweaks," he said. "We’ve gone the last two seasons with a way and tried to develop and get better at it.
"Nothing went to plan [against Liverpool]. I think the first chance we get, Fred has won the ball high up and plays a good pass through, great vision by Mason [Greenwood] to put Bruno [Fernandes] through. I think Fred had about seven or eight recoveries up there and it’s been very seldom that we’ve won the ball back against Liverpool as high up as we did.
"But, of course, being 2-0 down when they get the third goal because we’ve had to go for it at 2-0 and we were too open so… nothing went to plan."