Pogba has admitted he is looking to move this summer in search of a new challenge while agent Mino Raiola says he is trying to engineer his exit.
Real Madrid have been strongly linked with the World Cup winner but United are holding out for a fee of around £180million and the Spanish giants are unwilling to meet their demands.
Solskjaer recently said United have not received any official offers and provided fresh hope that he might yet stay after watching his side beat Perth Glory 2-0 in a pre-season match in Australia.
Asked if Pogba was being considered as a possible captain, he replied: “Yep. We’ve got quite a few possibilities there and when the season starts we’ll announce it.”
Ashley Young was handed the armband for the clash with an understrength Perth side which United won comfortably thanks to second-half goals from Marcus Rashford and James Garner.
Pogba started on the bench but Solskjaer changed every outfield player for the second half with Juan Mata taking over the captaincy from Young.
On who would be captain next term, Solskjaer added: “Of course, we’ll have a good assessment, see how pre-season goes.
“But, as you saw, Ashley Young will wear it when he’s playing until we decide on someone, yeah.”
Pogba laid on the assist for Rashford’s goal and gave a cryptic response when asked to comment on his future by reporters.
He said: “Who needs to talk? There’s no need.”
Solskjaer was also asked about potential incomings in the transfer window as he looks to straighten his squad that struggled badly at times last season.
He was understood to want all of his business concluded before their pre-season tour but recently confirmed is still looking to bring in new players who can provide the ‘X factor’.
Bringing in a new midfielder remains a priority even if Pogba does stay following the departure of Ander Herrera.
Bruno Fernandes, Sergej Milinkovic-Savic and Sean Longstaff are all reported targets, while they are also looking for a new centre-back.
Leicester’s Harry Maguire is at the top of their list of priorities but United are said to be put off by the Foxes asking price of around £85m.
And Solskjaer remained tight-lipped when quizzed on potential new faces.
“You need to speak to Ed [Woodward, executive vice-chairman] or others,” he said.
“We are working, as I said before. Timing is an issue but we are optimistic.”