The French midfielder did not start the derby clash after being dropped by United's Norwegian boss on Sunday and while he did come on at half-time, he was sent off 15 minutes into the second half for a high lunge on Liverpool's Naby Keita.
The Red Devils are still feeling the impact of the devastating defeat to their fiercest rivals, with Solskjaer's position as manager under threat - though United will keep the Norwegian in the role for now.
Paul Pogba (above) reportedly snubbed Manchester United boss Ole Gunnar Solskjaer after his red card against Liverpool
Solskjaer (above) is under threat after United were thrashed 5-0 by their rivals last week
And according to The Sun, Pogba apologised to his team-mates for his red card at Old Trafford on Sunday, though did not say anything to Solskjaer after the loss.
The reported snub raises questions over Pogba's future at Old Trafford, with the 28-year-old's contract at Manchester United expiring at the end of this season.
Pogba's agent Mino Raiola has insisted that there is 'no update' in regards to the midfielder's ongoing contract situation.
If Pogba fails to agree to an extension before January, he will be allowed to discuss terms and sign a pre-contract with a new club ahead of a move next summer.
Pogba's reckless tackle on Naby Keita (right) saw him sent off 15 minutes after coming on
The 29-year-old has been linked with moves to Real Madrid, Barcelona, Juventus and Paris Saint-Germain, while United are keen to keep the midfielder.
'You would have to ask Manchester United about it. At this moment everything is calm. There