British media have been awash with reports of van Gaal's demise, after a leak from an unknown source spread as Manchester United lifted the FA Cup on Saturday.
The Dutchman cryptically commented, "It's over" as he left the squad's Cup final hotel on Sunday, but it was not clear whether he was referring to his tenure or United's season.
The BBC said van Gaal had not been informed whether he was staying for next season, although it said Saturday's 2-1 win over Palace at Wembley was believed to have been his last game in charge of United.
There has been no official comment from the 20-time English champions, although the no-nonsense van Gaal was defiant after Saturday's Cup final.
"I show you the cup and I don't discuss it with my friends of the media, who already sacked me for six months," the 64-year-old said.
"Which manager can do that, what I have done?"
Despite winning their first trophy since the great Alex Ferguson's retirement in 2013, van Gaal's United didn't challenge for the Premier League title and failed to qualify for the Champions League.
Fans were also unhappy with the brand of football their team played under van Gaal.
United scored just 49 league goals this season, their lowest return for 26 years and many supporters felt van Gaal's approach ran counter to the club's traditional attacking style.
Mourinho saw his second spell as Chelsea manager end with the sack in December, after last season's champions made a miserable start to the their Premier League title defence.
Van Gaal arrived at United's Carrington training ground at approximately 8:45 am local time (0745 GMT) Monday.
He was followed into the ground by employment lawyer Paul Giltoy QC, who has long been retained by the League Managers' Association.
Among Gilroy's clients are David Moyes, whose sacking as United manager paved the way for van Gaal's appointment, Ferguson and England boss Roy Hodgson.
Van Gaal still has a season remaining on his three-year contract.
But United captain Wayne Rooney accepted this term's performance had not been good enough by the standards expected at Old Trafford.
"He has been under pressure," Rooney said of van Gaal.
"I think we all have,"
"That's part of playing for Manchester United and managing Manchester United - it is a massive club.
"You have to set standards and we fell below them this season.
"But we have to make sure as players we get back to those standards and make sure we are back challenging for the main trophies."
Meanwhile former United midfielder Ray Wilkins said Mourinho was the man to revitalise the club.
"He (Mourinho) is a winner - there is no two ways about it," Wilkins, also a former Chelsea assistant manager told talkSPORT radio on Monday.
"When he goes to a club, normally he wins a big trophy within the first couple of years."
Original source: Man United to make statement on Van Gaal