The pressure increased on Van Gaal on Saturday when United stretched their winless run to six matches with an embarrassing home defeat to Norwich.
The loss meant United dropped out of the top four in the Barclays Premier League and the club's supporters made their feelings known during the match and after the final whistle.
The home fans left inside Old Trafford booed their team off following the 2-1 defeat and some cried Mourinho's name during the game.
Mourinho is a free agent following his sacking by Chelsea and the Portuguese has reportedly made it known that he would be happy taking over at Old Trafford if a vacancy arises in the near future.
Mourinho has no plans to take a sabbatical and United are thought to be one of four big European clubs the 52-year-old would consider joining.
United refused to comment on speculation that Van Gaal has two games to save his job, but it is understood the Dutchman was at the club's training base in Carrington on Monday continuing his work as normal
The 64-year-old was also at the AON Training Complex on Sunday.
In the immediate aftermath of the defeat, Van Gaal insisted he had the support of United executive vice-chairman Ed Woodward and the players.
The United manager said: "Ed was very fast out of the dressing room but I feel that support, yes.
"The most important support is from the players because the players have to do it on the pitch and the chemistry between the players and the manager and his staff is very important."
Despite those words, Van Gaal must be cursing his luck that two of the best managers in football have become available during the most challenging period of his 17-month spell at the club.
Mourinho was keen on taking the United job when Sir Alex Ferguson retired in 2013, but the Scot chose David Moyes as his successor instead.
Ferguson and director Sir Bobby Charlton were known to have doubts about Mourinho, but they will have only a minor say in who succeeds Van Gaal should he be axed.
Woodward and the club's owners the Glazer family will bear that responsibility should the situation arise.
As well as Mourinho, Pep Guardiola will be available in the summer as he is leaving Bayern Munich.
United playmaker Juan Mata, meanwhile, insists he and his team-mates can arrest their slump over the next week when they face Stoke and Chelsea.
Addressing fans in his weekly blog, the Spaniard wrote: "We are really disappointed for not giving you what you deserve, because your support is amazing and unconditional, but I'm sure that we will turn this situation around."
Mata, who was sold by Mourinho to United for £37.1million in January 2014, was disappointed to lose to Norwich but hopes the Red Devils can return to winning ways at Stoke on Boxing Day.
"I was confident that we were going to get the three points on Saturday in Old Trafford, but that was not the case," said the 27-year-old, who will come up against his former club, now managed by Guus Hiddink, on December 28.
"We are in a complicated situation and in order to get out of it we have to stick together and believe in ourselves so we can get a positive result as soon as possible.
"We all feel frustrated, same as last week, and we need to win a game as quickly as possible because it will change our mood and give us confidence, I'm sure of that.
"Our next chance to do it will be on Boxing Day at Stoke, and that's not an easy game.
"No matter how tough the situation can be, I'm not even thinking about giving up.
"I believe that's the only way to get out of this bad streak: keep working hard and sooner or later we will get the results that we want."
Source : PA
Source: PA