Former Spurs boss Mauricio Pochettino has already been linked with replacing Solskjaer as United boss.
And according to the Sunday Mirror, Woodward has ‘privately insisted’ that he is prepared to remove Solskjaer from his position if United’s form worsens.
The report claims that Woodward uses Liverpool as an example of a club legend being sacked in the right circumstances after they fired Kenny Dalglish in 2012.
Solskjaer faces a tough task in the next few weeks as United face both Paris Saint-Germain and RB Leipzig in the Champions League, while they will host both Chelsea and Arsenal in the Premier League.
United are reportedly keeping tabs on Pochettino’s situation as the Argentine remains out of work since Spurs sacked him last November.
Speaking after United’s heavy defeat to Spurs, Solskjaer said: ‘It’s nowhere near good enough.
‘And when you have a defeat like this, which has happened at the club before, you’ve just got to look yourself in the mirror.
‘But we don’t see each other for ten days and that’s hard now because we need to batten down the hatches and get together because that wasn’t anywhere near good enough as a squad or a team.’