By Roberta Rampton

WASHINGTON (Reuters) - Ohio's Republican governor, John Kasich, took over the White House briefing lectern on Friday to praise the 12-nation Trans-Pacific Partnership, an unusual move that shows how hard President Barack Obama will push for the trade deal's approval.

Obama, a Democrat who is leaving office in four months, met with the governor, who fell short in his bid to be the Republican presidential candidate, in the White House Oval Office to discuss strategies for overcoming domestic political angst over the TPP.

The unlikely partnership comes as the White House makes a final full-court push to persuade Republican congressional leaders to approve the deal in a "lame duck" session after the Nov. 8 election. Both Republican and Democratic candidates have pilloried the TPP.

"We cannot get to the point in America that because a Democrat wants something, you can't agree with them," Kasich said in an impassioned plea to Republican lawmakers to back the deal, which is a key part of Obama's foreign policy legacy.

Kasich said the deal is vital for the U.S. economy and also to counter China and Russia, and said business leaders need to apply pressure on lawmakers to approve it.

"Right now, China is pushing hard to create their own trade agreement," Obama told reporters in the Oval Office ahead of the meeting, saying U.S. businesses were at risk of being "cut out" of Asia, the world's fastest-growing market.

"I promise you that China's not going to be setting up a bunch of rules that are going to be to the advantage of American companies and American businesses," Obama said.

Other business and political leaders also attended the meeting, including former Republican Treasury Secretary Henry Paulson, International Business Machines Corp Chief Executive Officer Virginia Rometty and former New York City Mayor Michael Bloomberg.

Republicans traditionally have backed free trade deals, but the party's presidential candidate, Donald Trump, has blamed the agreements for U.S. job losses and threatened to tear them up should he win.

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Senate Majority Leader Mitch McConnell has said the TPP would not get a Senate vote this year, and House of Representatives Speaker Paul Ryan has said he does not see enough votes for it to pass.

Democratic presidential candidate Hillary Clinton has also opposed the deal, which is unpopular with labor unions and environmental groups.

Obama has said he hopes opposition cools after the election. The White House has pointed to opinion polls showing most Americans support trade as a sign that the TPP could still squeak through Congress.

"If you're frustrated about rules of trade that disadvantage America, if you're frustrated about jobs being shipped overseas ... then you want to get this thing passed," Obama told reporters.

On his final trip to Asia as president earlier this month, Obama spent time reassuring nervous partners that the United States would finalize the TPP. But on Friday, Vietnam's parliament indicated it would not ratify the deal quickly.

White House spokesman Josh Earnest said Vietnam's decision was not a setback. "I think the real stumbling block, the real impediment, the obstacle here is (the U.S.) Congress," Earnest said, noting there was "every indication" that Vietnam would proceed as long as Congress approved the TPP.

(Reporting by Roberta Rampton; Additional reporting by Ayesha Rascoe; Editing by Jonathan Oatis and Leslie Adler)