The email from the former US ambassador to Australia is among 4,000 released amid a row about Mrs Clinton's use of private email for official business.
Mr Rudd's office said if such a request had been made, it was done without his authority.
Mr Rudd was replaced by his deputy, Julia Gillard, in 2010.
Part of the email, sent by ambassador Jeffrey Bleich to the State Department in June 2010, is blocked out.
"I spoke at length with former PM Rudd on Monday," Mr Bleich wrote.
He said Mr Rudd "did not raise the issue" but that his aide had later called "and noted that Rudd had not heard from S, [Secretary of State] and would have hoped for a sympathetic call".
"I have no strong point of view on this one. He has received such a call from POTUS [The President of the United States] already.
"But, I think he and S had a good relationship and he may want to talk to her about his future career goals."
A statement from Mr Rudd's office said he had made "zero request" for any call with Mrs Clinton.
"By that stage, Mr Rudd had already had a long conversation with President [Barack] Obama which was initiated by the President," it said.
Mrs Clinton's opponents have accused her of putting US security at risk by using an unsecured computer system for official business.
Mrs Clinton, who is seeking the Democratic nomination for the 2016 presidential election, says no classified information was sent or received on her personal email account, however, 125 emails were deemed confidential by the State Department.
She has said her decision to use a private email server at her New York home was a mistake.
Former Australia PM Kevin Rudd "hoped for a sympathetic call" from then US Secretary of State Hillary Clinton after he was ousted, an email says.