His claims to a full-time contract have been enhanced, though, by a remarkable 3-1 victory over PSG which has booked United a quarter-final berth in Europe.
Solskjaer has now overseen 14 victories across 17 games at the helm, with positive progress being made at home and abroad.
Shaw admits that it would be a surprise to see the 1999 Treble winner overlooked when a decision is made on United’s next boss, telling Sky Sports: "I don't think I need to say much. After all these results, it looks certain.
"We all love Ole and we love the job that he has done so far.
"We have had a couple of blips but I think it is nine away games now [that we have won] and they have not been easy ones, so that shows what a job Ole and his staff have done.
"Long may that continue."
Ashley Young has echoed the thoughts of his fellow United defender, with the long-serving Red Devils star suggesting that the club’s board have an easy decision to make.
He said of Solskjaer’s future after a thrilling night in Paris: "That is not down to the players to decide.
"If you look from the outside, you would say 'why not?' - he has been fantastic since he has come in.
"He has given everyone a breath of fresh of air and I think the way he has gone about the job has been unbelievable and the achievements we have made in a short space of time have been fantastic.
"Luckily for us as players, we do not have to decide. It is down to the owners and directors to decide that."