The Cumbrians sacked Greg Abbott earlier this month and former Stoke and Republic of Ireland midfielder Kavanagh, who has spent the last four years as a coach at Brunton Park, has steered the club to back-to-back victories since taking temporary charge.
The club's directors are holding interviews for the vacancy this week, while managing director John Nixon said after last week's win against Sheffield United that Kavanagh was high on their list of candidates.
"Like I say, the directors will make their decision and I am part of the process," Kavanagh said after Saturday's 3-1 success at Stevenage.
"I would love to carry it on, but there won't be any sulking if it doesn't go that way.
"Whatever happens will be for the good of the football club and that's always the most important thing.
"We've got another full week planned with games we want to go and see and with the training sessions to keep things bright and fresh.
"We've been asked to do a job and that's what we will do until we're told one way or another.
"I'm really enjoying the job at the moment, and I know I keep saying this, but all of the staff have been brilliant.
"I'll just continue doing the job as well as I can and we'll see where it takes us.
"I've made no secret about the fact that I want to take the club forward, but we'll see what the directors want to do.
"I'll just keep doing the best I can and, to be honest, I can't commit myself any more than that."