The pressure increased on Van Gaal when United stretched their winless run to six matches with an embarrassing home defeat to Norwich.
United's latest loss saw them drop out of the top four in the 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 after the 2-1 defeat and some cried Mourinho's name during the game.
Mourinho is a free agent after his sacking by Chelsea and the Portuguese has made it known he would be happy taking over at Old Trafford if a vacancy arises in the near future.
Although he is expected to fly to Portugal to spend some time with his family this week, 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 continuing his work as normal. The 64-year-old was also at the AON Training Complex a day after the loss to the Canaries.
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 as he is leaving Bayern Munich - reportedly to take over from Manuel Pellegrini at Manchester City.