It is understood all the club’s senior figures, including owners the Glazers, have discussed his future.
The writing was on the wall for Solskjaer when Watford flew into a 2-0 lead through Josh King and Ismaila Sarr.
Substitute Donny van de Beek pulled one back but United’s under-fire captain Harry Maguire was sent off after 69 minutes and late goals from Joao Pedro and Emmanuel Dennis piled on the misery.
Former United striker Solskjaer was appointed caretaker boss in December 2018 to replace Jose Mourinho.
The Norwegian became the first United manager to win his first six league games, beating a record held by Sir Matt Busby, and was handed the permanent role the following March.
United finished second behind Manchester City last season but currently lie seventh, six points off West Ham in fourth and 12 behind leaders Chelsea.