They report that United do not believe there is a viable candidate internally to take over as caretaker.
Antonio Conte wants the role but they are unsure whether he is the right man to take them forward.
The club also does not like the idea of making a change this early in a season and bringing in a replacement who may demand changes to the coaching staff.
Solskjaer may have his former boss, Sir Alex Ferguson, to thank for buying him more time.
Ferguson looked dejected during United's crushing 5-0 defeat to Liverpool at Old Trafford last weekend.
The United legend was in the stands to watch his old side play their fierce rivals.
The camera panned to him after Mo Salah scored Liverpool's fifth just after half-time and to say he looked fed up would be an understatement.
However, despite not looking happy on Sunday, Ferguson retained his faith in his ex-player.
According to journalist Fabrizio Romano, Ferguson pushed for United to retain Solskjaer for now.
Ferguson believes the Norwegian should be given the chance to turn the club's fortunes around.
Per Samuel Luckhurst of the Manchester Evening News, Ferguson travelled to United's training ground, Carrington, on Tuesday morning for talks.
Man United Fan Terry EXPLODES as Solskjaer ISN'T SACKED (5 minute watch)
Romano adds that Ferguson is 'defending' and 'protecting' Solskjaer from losing his job.
It must be reassuring for Solskjaer that his former manager and one of the best the game has ever seen is still backing him to do well.
It appears that Solskjaer will now be in charge for United's game against Tottenham on the weekend and also against Atalanta and Manchester City.
Whether the 48-year-old can use the time that Ferguson and his other supporters have bought him remains to be seen.