Sunday's 2-2 home draw with Aston Villa heaped more pressure on Solskjaer with United down in ninth place.
The Sun says afterwards Solskjaer made it clear to the players how precarious his position was, with ex-boss Jose Mourinho's Tottenham the visitors tomorrow and the derby with Pep Guardiola's Manchester City at the Etihad on Saturday.
A United source said: “Ole was a little bit emotional and told them that if they don't get two results this week, he could be gone.
"It is the worst time to be facing Spurs with Jose coming back."
The figure of Mauricio Pochettino waiting in the wings after his sacking by Spurs has increased the pressure on Solskjaer.