A 5-0 defeat at home to Liverpool on Monday (AEDT) was the lowest point of the Norwegian's time in charge, with a third loss of the Premier League season leaving United eight points behind early leader Chelsea.
His first-team squad reassembled at the club's Carrington training base two days on from that debacle, with Solskjaer still at the helm. He reportedly arrived early in the morning to put his players through their paces.
Senior United officials are said to have held a meeting on Tuesday (AEDT), with executive vice-chairman Ed Woodward joined by managing director Richard Arnold and co-chairman Joel Glazer.
United has repeatedly backed the Norwegian, who took over from Jose Mourinho in December 2018, initially as an interim appointment. It considers the former United striker an ideal figurehead, and Solskjaer remains a popular figure among supporters, even with those who want a change of manager.
The Guardian reported that Solskjaer still retains the support of Woodward, Arnold and United's influential former manager Alex Ferguson, with Joel Glazer set to have the final say. Similarly, The Telegraph said Solskjaer has been given three games to show he should be retained.
United would be wary of removing Solskjaer if there is no suitable candidate available to replace the 48-year-old, and early signs point to him likely remaining in charge for the Premier League clash with Tottenham on Sunday (AEDT). That is followed by games against Atalanta in the UEFA Champions League before a huge Premier League derby with Manchester City.
Former Chelsea, Inter Milan, Juventus and Italy boss Antonio Conte has been linked as a possible successor and is said to be open to the idea. Zinedine Zidane, however, would not be interested in the job, according to reports in Spain, where he had two spells in charge of Real Madrid.
Solskjaer's disappointment after the Old Trafford defeat was plain, as he said: "I have never felt any worse than this. This is the lowest I have been. I accept the responsibility. That is mine today and it is mine going forward.
"I do believe in myself and I am getting close to what we want at the club. The results lately haven't been good enough. Hands up. Next week is Tottenham away. Then Atalanta. Then Man City. We have to go into this with the right frame of mind."
In July, Solskjaer signed a new three-year contract at United, and the club backed him as Cristiano Ronaldo, Jadon Sancho and Raphael Varane joined during the transfer window.
He has an average points-per-game record of 1.83, which is the third highest in United's Premier League history. In that respect he only trails his predecessor Mourinho (1.89) and the long-serving Ferguson (2.16).
Solskjaer has a 51.89 per cent win record from his 106 Premier League games in charge of the Red Devils to date.