The England and Wales Cricket Board (ECB) is creating the job as part of a restructure after the departure of managing director Paul Downton.
Vaughan, 40, said he was "more than open" to the role.
Stewart, who played a record 133 Tests, said: "If the ECB want to speak to me, of course I'll speak to them."
But the Surrey director of cricket, 52, added that the ECB would have to seek permission from his county first.
"You'd be silly not to listen," said Stewart. "That's not just me, that's anyone involved in the game who feels they can make a difference at the top level."
Vaughan won more Tests - 26 - than any other England captain and scored 5,719 Test runs at an average of 41.44.
"I am certainly one of the few who would have the phone ready to have a conversation, whether in this role, another role, or a forward-thinking visionary trying to move English cricket forward," he said.
Downton, 58, left his post on Wednesday, a month after England were knocked out of the World Cup in the group stages.
He was also involved in the sacking of former England captain Kevin Pietersen in February 2014.
Downton's departure came on the day Wisden editor Lawrence Booth accused the ECB of having "repeatedly lost touch", citing its handling of the Pietersen affair, concerns with the decline of the Test and one-day sides, and a fall in grassroots participation.
ECB chief executive Tom Harrison said Downton's departure was about being "accountable for reaching the standards we aspire to".
He added that the new role would have "a clear focus on delivering a world-class performance environment for all formats".
Vaughan, a fan of Harrison, said: "English cricket has to look deeper than just removing one person. Structurally the game has to change. Removing Paul Downton is not going to turn it around overnight."
Former England bowler Steve Harmison backed Vaughan to come up with a plan that would ensure England "are a massive force in world cricket".
But ex-England all-rounder Derek Pringle said the role of guiding England's fortunes had become "a bit of a poisoned chalice".
Former England captains Michael Vaughan and Alec Stewart say they are interested in becoming the country's new director of cricket.