While two wins on the bounce has eased the pressure on the Dutchman slightly, Van Gaal's position remains precarious, as United are far from guaranteed UEFA Champions League football next season, per James Ducker of the Times.
Following his sacking by Chelsea in December, Mourinho has long been considered the favourite to take over at Old Trafford should Van Gaal depart, and Sky Italia journalist Alessandro Alciato now reports a deal is "close" (via James Robson in the Manchester Evening News):
Alessandro Alciato @AAlciato
@SkySport Mourinho-Manchester United: accordo vicinissimo. Che derby con il City di Guardiola!2016-2-3 19:30:22
Van Gaal's three-year contract with United does not run out until the end of the 2016-17 campaign, and the Old Trafford hierarchy would ideally like him to see it through to the end, per Robson.
They have kept faith with him through various rocky patches this season, including United's late-2015 run of eight matches without a win, during which they were knocked out of the Champions League at the group stage.
Not only have results been poor this season, but Van Gaal has been widely maligned for the boring style of play he has introduced at Old Trafford, per the Guardian's Jonathan Wilson.
A 3-1 FA Cup win against Derby County and a 3-0 defeat of Stoke City have recently spawned some greater optimism among the United faithful, with the performance against the Potters particularly heartening in an attacking sense, per Bleacher Report's Dean Jones:
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However, the Red Devils remain five points off the ever-important fourth spot in the Premier League, leaving their Champions League ambitions for next season in grave danger.
Should results not continue to improve and United's top-four prospects become less and less likely, the Old Trafford top brass could yet pull the trigger on Van Gaal, paving the way for Mourinho's instalment as manager.
Van Gaal could befall a similar fate if he is unable to force United into the top four, and the spectre of Mourinho will pile on the pressure.
Further calls for the former Porto, Real Madrid and Inter Milan boss' instalment at Old Trafford were sparked by the recent confirmation that Pep Guardiola will take over at Manchester City in the summer, per BT Sport's Ian Darke:
Ian Darke @IanDarke
Mourinho might seem more appealing to Utd today. Feisty pragmatism to harry and hustle Pep at every turn in Manchester ...in theory!2016-2-2 12:02:37
The pair have gone head to head in the past in Spain when Mourinho was at Real and Guardiola at Barcelona.
The prevailing wisdom now suggests that hiring Mourinho would be the ultimate response to City's high-profile appointment, per MailOnline's Joe Bernstein.
However, United have stuck with Van Gaal this far, and if he can engineer a strong end to the season and qualification for the Champions League, he could yet remain in the top job despite the looming presence of Mourinho.