The 2-0 loss at the windy Britannia Stadium pushed United down to sixth place in the English Premier League, and Van Gaal is sensing the pressure after barely 18 months in charge.
"I can also quit by myself," Van Gaal said when asked if he had received assurances about his future from the club hierarchy. "That is something I speak to (vice chairman) Ed Woodward (about) by himself — not with you (reporters).
"It is not always ... the club has to fire or sack me. Sometimes I do it by myself. I am the one who wants to speak first with the board of Manchester United and with my members of staff and players — and not with you."
United was in disarray defensively in the first half when Memphis Depay's horrendous defending allowed Bojan Krkic to score, and Marko Arnautovic powered in Stoke's second goal from outside the penalty area.
United captain Wayne Rooney watched it all from the sidelines after being dropped to the bench for the first time in a league match in almost two years. United came to life only in the second half after Rooney was brought on, but Stoke goalkeeper Jack Butland thwarted any hopes of a comeback.
"The circumstances were difficult — not only the circumstances, the wind, but also the pressure," Van Gaal said. "That is in my opinion the reason they don't dare to play football (in the first half)."
The losing streak started with Champions League elimination, and has also seen United drop out of the Premier League top four by losing to Bournemouth and Norwich.
Winless in seven matches, United has little time to recover its confidence, with a home match against Chelsea on Monday.
Asked if he would be in charge for the game, Van Gaal said: "You will have to wait and see. But I think so."
"I try to do everything but the pressure shall be every match higher and higher, and we have to solve that problem," Van Gaal, who has been in charge since July 2014, added. "It's more difficult because I am now part of the four matches we have lost and people are looking at me."
A day after British astronaut Tim Peake displayed a Stoke flag on the International Space Station, the central England team was in a different orbit to United in the first half.
Depay's mind seemed to be on another planet when he sent a downward header back toward goalkeeper David de Gea in an attempt to clear Geoff Cameron's deep ball.
Depay, a winger in the right back position, was unaware that Glen Johnson was primed to dart in and seize possession before crossing for Krkic, who took a touch then slotted the ball into the net in the 19th minute.
"It was a bad goal to give away," Van Gaal said.
If United's players were fighting to save their manager's job, they didn't demonstrate it. Their response lacked any burning desire to end their slump.
And the mission facing United became even tougher seven minutes later. Krkic's free kick was blocked by the United wall but the ball came back out to the edge of the penalty area and Arnautovic's powerful strike beat De Gea.
United was offering little in response, with Depay's free kick parried by Butland, and Marouane Fellaini was offside as he tried to react.
United was spared further misery before the interval when Arnautovic ran onto Krkic's throughball but missed the target with his shot.
As usual, on the United bench there was little outgoing evidence of concern, with neither Van Gaal nor assistant Ryan Giggs springing out of their seats to berate or inspire their team.
Van Gaal's about-turn came at halftime when Rooney spent the break warming up before coming on to replace Depay.
Rooney's first significant contribution came after 20 minutes on the field, cutting the ball back for Fellaini six yards in front of goal. But rather than slamming the ball high into the net, Fellaini shot low and lacking power, allowing Butland to make a diving save.
In the closing minutes, Rooney this time set up Juan Mata but again Butland made the vital save. And Butland also pushed Anthony Martial's attempt round the post as Stoke completed a December double over Manchester clubs, having already beaten City.
Stoke is three points behind United, which manager Mark Hughes used to play for. United is three points adrift of fourth-place Tottenham.
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