McDermott parted company with United on Friday after a 10-month spell with the Yorkshire outfit.
Kelly, who enjoyed a 15-year playing career with the Whites, is "more than confident" that he could lead the club into the playoff places in the Championship table should he be offered the opportunity.
"The club needs someone who has the club, fans and city at heart," he told Sky Sports News.
"It deserves someone with that passion and desire to succeed to get back to being one of the biggest clubs in the country, exactly where the the fans, club and city belong.
"This season, looking at the current predicament, I am more than confident I can take them to the play-offs, given the right backing and support. I believe I have all the credentials to do this and more."
The former Republic of Ireland international has no previous professional managerial experience since retiring from the game.