United have lost two of their first three Premier League fixtures so far.
Solskjaer’s men went into the international break at the back of a 6-1 hammering from Tottenham last Sunday.
Former Spurs boss, Mauricio Pochettino, is already being linked with replacing Solskjaer at Old Trafford.
It is understood that Woodward has, “privately insisted” he is prepared to remove Solskjaer from his position if United’s form worsens.
Woodward will use the example of Liverpool sacking a club legend in the right circumstances, after they fired Kenny Dalglish in 2012.
Solskjaer faces a tough task in the next few weeks, as United play both Paris Saint-Germain and RB Leipzig in the Champions League, while they will host both Chelsea and Arsenal in the Premier League.
