The Daily Mail says United's board have no intention of getting rid of him, even though the club have their lowest points tally after 24 games in over 30 years.
United could have moved to within three points of fourth-placed Chelsea with a victory on Wednesday night, but instead find themselves six points behind Frank Lampard's men and level on points with Tottenham and Wolves.
Their total of 34 points is the lowest since the 1989-90 season, when they had 25 points from 24 games and finished the campaign down in 13th position.
Wednesday night's result meant Solskjaer quickly became the bookmakers' favourite to be the next Premier League manager to lose his job.
However, club owners the Glazers and executive vice-chairman Ed Woodward have no plans to axe the former Cardiff boss, despite their hopes of reaching next season's Champions League suffering another blow.