The 38-year-old has been in interim charge of Killie for their last four games following Lee Clark's move to Bury.
During that time, McCulloch's Kilmarnock have won two and lost two.
"I'm enjoying it just now so there's no reason to start asking for a longer term," McCulloch said.
"I'm enjoying it day-to-day, I'm enjoying it week-to-week. I think it'd be a little bit selfish of me to say, 'give me it until the end of the season'.
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"I don't think anybody can really do that in my position.
"We had a good result last week against Ross County and I got a text from a couple of the board saying 'great result', 'keep it going'. I'm happy with that."
That 2-1 win over County took Killie up to seventh in the Premiership and featured a winning goal from Kris Boyd, who has now scored 100 league goals for two clubs in Scotland's top flight.
Boyd is in his third spell at Rugby Park and McCulloch said of his former Rangers team-mate: "I've had the privilege of playing next to him.
"His goal record's phenomenal. He's a winner. He's great about the dressing room for the kids. The scary thing is there's still more goals in him so I wouldn't rule him out breaking more records.
"He doesn't get anywhere near as much credit as he should get. He's had stick, basically, from all fans in Scotland and he just manages to keep popping up with goals."
Kilmarnock interim manager Lee McCulloch says it would be "selfish" of him to ask for the job on a more permanent basis.