Pogba has been out of action for Manchester United since injuring his hamstring against FC Basel on September 12.
Reports have suggested the France international is set for a long time out with the Red Devils mulling over whether to have the player undergo surgery.
However, the Independent say United are hopeful of having Pogba back in their ranks this month.
It is said the Red Devils are targeting their League Cup game against Swansea on October 24 for Pogba to return.
Manchester United were supposedly ready to send the 24-year-old to a hamstring specialist.
But it appears Jose Mourinho has now opted against doing so.
The Independent add that, if Pogba returns for Swansea, then he will be in contention to start their Premier League clash against Tottenham four days later.
Speaking in September, Mourinho voiced concerns that the injury would be ‘long term.’
“I have the hope to see them in training and to have a positive answer, so long-term injuries I don’t speak about them,” he said.
“So Ibra [Zlatan Ibrahimovic] and Pogba and Marcos Rojo, these players I don’t think about them.”
United have coped well in Pogba’s absence, winning their last five games across all competitions.
However, prior to his injury, the former Juventus star was hitting his stride.
Pogba found the net in games against Swansea and West Ham.
And he also starred in a 2-0 victory against Leicester, with injury robbing him of the chance to build on his promising start to the campaign.