The Norwegian has been placed in interim charge at Old Trafford following Jose Mourinho's exit, but Wednesday night's 2-0 win at Newcastle extended his perfect start to four wins from four, equalling the record set by Sir Matt Busby.
Asked if he would not want to leave in May, he said with a smile: "Of course I won't, I don't want to.
"It's such a great bunch of players, a fantastic atmosphere, but it's the next game, it's the next game, it's the next game and I'm doing my job as long as I'm here."
Second-half goals from substitute Romelu Lukaku, who scored with his first touch after Magpies keeper Martin Dubravka spilled Marcus Rashford's free-kick, and Rashford were enough to take the points at St James' Park.
However, Solskjaer was in no mood to rest on his laurels after matching the feat of his famous predecessor, remembering his United manager Sir Alex Ferguson's approach to winning runs.
He said: "That will be in the books, but it's nothing that I'm thinking of. I'm just thinking about the next game because if you win four, you can win another four at this club. That's the challenge and that's the standard that we're known for.
"The gaffer used to challenge us on them and of course when we've won four, you can go onto the next four and think about them."