Chief selector Marsh is set to step down when his contract runs out in mid-2017, and Gillespie said he was "open to offers" after leaving English county side Yorkshire.
"There is a national selectors' job up next year and I might put my name up for that," he told the Adelaide Advertiser.
"All I have done the last five years is select teams for Yorkshire. In county cricket you are always selecting teams. The buck stops with you. I wouldn't rule anything out."
Gillespie is now back in Australia for family reasons and will coach the Adelaide Strikers in the summer's Big Bash League after a hugely successful five-year spell at Headingley.
He was strongly linked with England's vacant head coach position last year and is seen as a potential successor to Lehmann.
"It was a tough decision to leave Yorkshire but the time was right for family and my career moving on," he said.
"I will put my energy into the Big Bash with the Strikers, see what happens. I guess I am open to offers.
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