The Daily Mail says the 46-year-old Norwegian and his No 2 Mike Phelan have yet to receive confirmation of United's decision as they look to plan ahead for next season.
The same applies to interim boss Solskjaer's assistant manager Mike Phelan.
It's understood that Phelan will return to Australia during next week's international break in his role as sporting director for Central Coast Mariners.
Meanwhile, Solskjaer has repeatedly answered questions about his future by saying that he will continue to focus on his job until United make a decision, and that genuinely seems to be the case despite a clamour for his appointment from players and fans.
He has been involved in long-term planning over the club's summer tour - which begins in Perth in Australia in July and is also likely to take in Singapore and China - as well discussions about possible transfers.