Woodson, 60, coached the Knicks from midway through the 2011-12 season through 2014. He went 109-79 in his two-and-a-half seasons.
"I don't want to hide the fact I'd love to be back," Woodson told Marc Berman of the New York Post. "I'd like to finish what I started. At the end of the day, you want to come to New York, based on my body of work there. I want it to be mutual. I want them to want me. I hope they call me."
The Knicks fired Jeff Hornacek last week after he went 60-104 in his two seasons on the bench. Mike D'Antoni is the only Knicks coach this century to last three full seasons. Woodson is the only coach this century aside from Jeff Van Gundy, who was hired in 1996 and stayed until being fired in 2001, with a winning record.
New York made the playoffs twice under Woodson, and he led the Atlanta Hawks to three postseason appearances in his six years with the franchise.
"I would be crazy not to," Woodson said when asked if he wanted to be a head coach again. "I'm still under contract with the Clippers. Somebody calls me and gives me the opportunity to talk, I'd be happy to talk."
The Knicks are also set to interview former Memphis Grizzlies coach David Fizdale and former NBA guard Jerry Stackhouse, who coaches Toronto's G League team, for their opening. The odds remain against Woodson getting the job, but he does have an existing relationship with president Steve Mills from his first tenure.
The Knicks have not posted a winning record or made the playoffs since Woodson's exit.