McTominay is expected to be sidelined for two months with a knee ligament injury while Pogba needs surgery on an ankle problem and could be out for a month.
The United board will back Solskjaer with funds to bolster his squad, but only if long-term targets become available.
Longstaff was an option in the summer but Newcastle's valuation of £50 million proved too big a stumbling block. Can is out of favour at Juventus and keen to leave but United face competition from Borussia Dortmund.
James Maddison remains Solskjaer's top midfield target but there is acceptance Leicester will not want to do business midway through a season during which they are challenging for a place in the top four.
Aston Villa's Jack Grealish and West Ham's Declan Rice are also being monitored. Nemanja Matic is open to leaving Old Trafford in January after falling down the pecking order but it is unlikely he will be allowed to depart while Solskjaer is so short of bodies in midfield.
"Of course, it's playing on our decision-making when you've got two players out -- Paul out for a month and Scott for two," said Solskjaer after the defeat at the Emirates on New Year's Day.
"That will give us less options and we've got loads of games and important games. Still, it's got to be right and that's the point here.
"If he's available and it's possible, we'll do something, definitely. We're looking at one or two."
Meanwhile, United have moved to clarify comments made by Solskjaer that it was Pogba's "people" who decided he needed an operation.
Sources have told ESPN FC that scans carried out by the club's medical staff confirmed the 26-year-old would need an operation to resolve an ongoing foot problem and when the results were shared with Pogba's own specialist, it was agreed the Frenchman will have to undergo surgery "as soon as possible."