The France international cost Manchester United two points at Molineux Stadium on Monday night following his second-half penalty miss.
Anthony Martial broke the deadlock in the 27th minute with a clinical finish after Marcus Rashford teed up the France international.
However, Wolves were level in the second half thanks to Ruben Neves’ brilliant strike that left David De Gea with no chance.
Pogba manufactured a penalty when the World Cup winner’s tenacious run was illegally halted by Wolves defender Conor Coady.
After a brief discussion with Marcus Rashford about the penalty, Pogba took the spot kick but the French midfielder’s effort was saved.
The England striker scored a penalty in Manchester United’s 4-0 victory over Chelsea FC last weekend to raise eyebrows about Pogba’s decision to take the spot-kick.
Former Liverpool FC defender Carragher was quick to suggest Solskjaer should remove Pogba from penalty duty.
“That will never work,” Carragher told Monday Night Football on Sky Sports. “You will get that on pitch conversation every time United get a penalty.
“You don’t need that conversation. The manager needs to be stronger.
“Or Paul Pogba should think of the team and say Rashford is the best penalty taker. It’s cost them two points.”
Manchester United finished in sixth place in the Premier League table last season to miss out on a spot in the Champions League.
Solskjaer signed Swansea winger Daniel James, Crystal Palace defender Aaron Wan-Bissaka and Leicester City defender Harry Maguire in the summer transfer window.
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