Washington (CNN) Ohio Gov. John Kasich says he feels it's his "responsibility and duty as a leader" -- no matter the political cost -- to help President Barack Obama shepherd the Trans-Pacific Partnership through Congress.

"I have never been an ideological supporter of free trade. The ideologues use to come to me and be frustrated with me," he told the media during the White House press briefing.

"But when you look at these agreements in a real sense - and this one is much different than even NAFTA," Kasich added. "This is China. This is Russia. These are fledgling countries in Asia and we want to pivot to Asia? We have to do this."

The two-term Republican governor met with Obama in the Oval Office on Friday. In an exclusive interview Thursday night, he told CNN's Dana Bash he doesn't mind the political backlash he could face.

"I welcome the fact that people will criticize me for putting my country ahead of my party," Kasich said.

Kasich and Obama could be facing an uphill battle: Both Donald Trump and Hillary Clinton oppose the 12-nation Pacific Rim deal, which Obama has pitched as a way to counterbalance China's rise in the region.

"This is the first time the candidates in both major political parties say they are opposed to free trade. It's astounding to me," Kasich said.

In the Oval Office, Kasich joined Obama, former New York City Mayor Michael Bloomberg, former George W. Bush administration Treasury Secretary Hank Paulson, Atlanta Mayor Kasim Reed and others for a meeting on the 12-nation Pacific Rim deal.

"This is an opportunity for the Congress to carry out its responsibility," Kasich said.

"You gotta get this done. I'd love to think it could happen next year. I'm not convinced it can happen after this year," he added.

Kasich waded into the presidential debate a bit, but mainly focused his efforts on rallying Republicans away from Trump's protectionist stance and toward the party's pro-trade orthodoxy.

"I think I need to spend my time making the case that we don't want to hurt United States national security issue(s), we don't want to turn our back over there, and frankly, we don't want to put ourselves in a position where we're not taking advantage of economic opportunities," he said.

Kasich also defended the prospect of Obama pushing the TPP toward passage in a "lame duck" session of Congress, after the November 8 election but before a new president and Congress are sworn in.

"Frankly, if I have to come down here and spend some time lobbying my Republican colleagues, I'm more than glad to do that," Kasich said.

"There's definitely some people I can call and talk to," he said. "This is a big deal. I mean, if we were to just walk away with this -- with both candidates saying they don't want this -- we turn our backs on Asia."

He also played down the political potency of Trump's anti-trade position in manufacturing-heavy Ohio, saying it's not why Trump might win the state.

"I don't think that's what it will turn on. And frankly, part of the problem that the Clinton campaign has is that there's literally no enthusiasm for Hillary Clinton. There just isn't right now. I don't know if it will change but that is her challenge," Kasich said.

Kasich also talked politics, saying it's "very unlikely" he will vote for Trump, his former GOP primary rival.

"Too much water under the bridge," Kasich said.

But one other thing's for sure: "I'm not voting for Hillary."

Referencing his refusal to attend the Republican National Convention in his home state, Kasich said, "I think my actions have spoken very loudly -- louder than even my words."

Touting his campaign stops for House and Senate Republican candidates across the country, he said his goal this fall is to "get Republicans out to vote -- period."

"We want to re-elect Senator [Rob] Portman and we want to elect people down-ticket," he said, conceding he knows that'll help Trump, as well.

"Because of the success that we've had in Ohio, Ohio's turning more red. It's more Republican," he said. "And why? Things are going pretty well. So when you produce things and people feel good, it has an impact. I think Ohio now is more Republican than it's been in a very long time, and that, of course, accrues to any Republican."

If Trump wins the election and offered him a role in his administration, Kasich said he'd help -- as he would with any president.

"If there's something I can do and something I agree on, absolutely," Kasich said.

As for his own political future -- and whether his aggressive campaign schedule for Republicans this cycle foreshadows a 2020 run for president -- Kasich deflected.

He said he's campaigning for House and Senate candidates "because I feel responsibility."

"I mean, you know, look, I don't have any idea what I'm going to do with my political future, I really don't," he said. "I'm not closing any doors. But I'm not here plotting and scheming what's going to happen down the road."