Solskjaer is looking for a new skipper after the departure of Antonio Valencia at the end of season.
Manchester United have made two signings so far this season, both young, promising talents.
Targetting a fallback option, the Premier League giants reportedly now have set their sights on the Fiorentina youngster who also was a target for Jose Mourinho but United failed to get him in January.
Last season ended with just two wins from the final 12 games and Solskjaer is keen to get off to a flying start when the new campaign kicks off against Chelsea on August 11.
An under-pressure United struggled in their pre-season opener against a weakened Perth Glory, who won Australia's A-League Premiers Plate last season.
"There are players here that won't be part of that", the coach told Sky Sports after their hefty defeat to Everton.
Let's see when the season starts.
The youngster joined from Swansea to become Ole Gunnar Solskjaer's first signing as manager and instantly became a hit with some sections of supporters. "Any player who plays against, say, Marcus Rashford, Anthony Martial, Dan James, with the pace they've got, I think you're anxious that they can hurt you - even if they have an off day".
Asked directly whether Pogba is one of the names under consideration, Solskjaer replied: "Yep".
The United manager was also happy to discuss the impact that his two latest signings would have on the team, highlighting how the pace of Aaron Wan-Bissaka and Daniel James would bring another dimension to his squad.