Britain's Guardian newspaper earlier reported that the club's executive vice-chairman, Ed Woodward, had persuaded the Dutchman to stay following the 1-0 defeat which resulted in fans booing the team and Van Gaal off the pitch.
Woodward told Van Gaal to speak to his family before making a decision over his future, the newspaper said. The Dutchman flew to the Netherlands on Sunday for his daughter's birthday.
Van Gaal is expected to hold fresh talks with United's hierarchy on Tuesday when he takes first-team training.
The loss to Southampton left United in fifth place in the league table, trailing leaders Leicester City by 10 points and five points adrift of fourth-placed Tottenham Hotspur.
The defeat also marked the first time this season that the increasingly impatient fans showed open dissent against the manager.
Van Gaal admitted for the first time after the game that he had failed at Old Trafford, saying he was disappointed and frustrated that he had not lived up to the fans' expectations.
(Reporting by Ossian Shine; Editing by Jon Boyle)
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