United slipped to a demoralising 2-0 defeat at the London Stadium, stretching their winless run on the road to seven Premier League matches.
West Ham winger Andriy Yarmolenko landed the first blow a minute before half-time and full-back Aaron Cresswell supplied the knockout punch with a spectacular late free-kick.
The Red Devils have won only five of the 17 matches played since Solskjaer was given the managerial job on a full-time basis in March.
Despite that disappointing run of form, the Daily Mail claims that United believe the Norwegian is "still the right man for the job" at Old Trafford.
United's hierarchy "accept that it will take time for him to turn the club's fortunes around" and they "will back Solskjaer in the January transfer window if he wants to strengthen his squad again".
United's striking department looks thin on the ground as the club allowed Alexis Sanchez and Romelu Lukaku leave in the summer without any replacements coming in.
Recent injuries to Marcus Rashford and Anthony Martial has highlighted the lack of depth in that position, with youngster Mason Greenwood missing the West Ham match with tonsillitis.