That statement, along with a call by Christie to cut Medicare benefits, was captured on tapes from an undisclosed trip that Christie made to Colorado on June 28. It was published online by the left-leaning magazine Mother Jones.
The event, a fund-raising and strategy session at which the governor was a keynote speaker, was sponsored by the controversial conservative financiers David and Charles Koch.
In an interview Wednesday, Oliver called the governor's statements about her "outright lies" and said, "At no time did I ever, ever pick up the telephone, call Gov. Christie and ask him to 'save my leadership.' "
She said the governor was engaged in a "chest-thumping bravado entertainment session" for Republican donors.
"That to me is a characterization of someone that is not fit to lead the state. . . . I think it's disgraceful," Oliver said.
A spokesman for Christie said the governor stood by his story.
Asked about a deal with Oliver at a news conference in Atlantic City, Christie described it as "another example of the way we've found to work together."
The trip to Colorado was not on Christie's public schedule. The state Republican Committee paid for his flight.
Such travels out of state are only likely to increase because the Republican Governors Association announced Wednesday that Christie had been elected vice chairman of the organization, boosting his fund-raising and party leadership responsibilities.
In June, Christie was introduced at the Ritz-Carlton Beaver Creek Resort by David Koch (pronounced Coke), who with his brother owns the second-largest privately held company in the country, Koch Industries, an energy and chemical conglomerate.
The brothers have reportedly spent more than $100 million supporting conservative causes, including Americans for Prosperity, which helps bankroll the tea-party movement, and are adamant opponents of the idea that human activity causes global warming.
David Koch said at the session that he and Christie met in his New York office five months earlier and spoke for two hours on problems in New Jersey government.
"At the end of our conversation I said to myself, 'I'm really impressed and inspired by this man.' He is my kind of guy," Koch said.
Americans for Prosperity has fought the Regional Greenhouse Gas Initiative (RGGI), a 10-state cap-and-trade agreement to reduce greenhouse gases.
In May, Christie announced that he would withdraw New Jersey from the pact, saying that while he believed humans cause climate change, he did not believe RGGI was effective in addressing the problem.
Business groups applauded the move, and environmental advocates said the withdrawal would set the state back.
On the tape, Koch praised the decision as attendees applauded.
"Another example of Gov. Christie's commitment to the free-enterprise system is that only a few weeks ago, he announced that New Jersey would be withdrawing from [RGGI], which would have raised energy costs, reduced economic growth, and led to very little, if any, benefit for the environment," Koch said.
Jeff Tittel, president of the New Jersey Sierra Club, said the tape "shows that the governor and the Koch brothers have the same political agenda."
Christie's speech did not touch on RGGI, though, and much of what he said was similar to what he has said to audiences of all political stripes in New Jersey - including doing an impression of former Gov. Jon S. Corzine.
At one point, however, Christie went further than usual in a call for the federal government to control spending.
"We're going to have to reduce Medicare benefits," he said. "We're going to have to reduce Medicaid benefits. We're going to have to raise the Social Security age. We're going to have to do these things. We're going to have to cut all types of other government programs that some people in this room might like."
He praised the audience, saying, "I'm here because it will be you - the people in this room are the modern-day patriots who will save this country or let it go by the wayside."
Christie talked at length about his efforts to solve New Jersey's financial problems, explaining how he pushed a controversial bill to overhaul public workers' health and pension benefits through the Democratic-controlled Legislature.
Recounting the hours before the vote in the 80-member Assembly, Christie said: "The reports I was getting out of there were not positive about what was going on to my friend the speaker. She was taking a beating at the hands of her own party."
Hundreds of angry union workers had gathered outside and in the gallery of the Assembly, deriding Democrats who supported the bill as traitors.
At 5:30 that evening, according to the governor, Oliver called and said to him, " 'I want to post the bill but I think when I go on the floor, my own party's going to take a run at me to remove me as speaker.' . . . She said, 'I think the only way I survive is if the 33 Republicans in the chamber will agree to vote for me for speaker. Can you work it out?' "
Christie said he went to the Republican Assembly caucus room and said, "Probably for the only time in my governorship I'm going to actually ask you to vote for a Democrat. . . . If they take a run at her on the floor, I need all of you to vote for her for speaker."
The Republicans, he said, looked at him in surprise, but agreed.
Assemblyman John Bramnick (R., Union) said that while there was Republican support for the governor and the speaker, he did not recall such a meeting.
A challenge to Oliver's leadership was rumored in June, as some Democrats fumed that the Assembly needed a leader who would more aggressively push back against Christie, but it never materialized. In question is whether she will have the votes to hold on to her post when her term is up in January.
Contact staff writer Maya Rao at 609-989-8990, firstname.lastname@example.org, or on Twitter @Mrao_Inquirer
Inquirer staff writer Jacqueline Urgo contributed to this article.