Cuomo, a Democrat, said Tuesday morning his focus would be to get Trump more involved in helping states with testing, especially in their native New York, hit hardest in the outbreak.
"Testing. What does testing mean and how can we do it? How can the federal government work in partnership with the states?" Cuomo said at his daily news conference in Albany.
Trump, responding to criticism from Cuomo and other governors -- both Democratic and Republican -- about a lack of coordination and supplies, has resisted a bigger federal role, saying testing is a job for the states.
"It is a situation where however you do it, it is going to be a blame game afterwards. I heard the president in his briefing yesterday. He is right. This is one of those thankless tasks," Cuomo said. "Everyone will say, ‘You did not do enough.' I get the instinct to distance yourself from it."
"But it is a situation where you need everyone to work together and to understand quickly who is in a better position to do what. From my point of view, I think the federal government needs to take the national manufacturer supply chain issue," he continued.
Asked Monday night the reason for the visit, Trump responded, "I don't know. He wanted to. We, believe it or not, we get along."
As he headed to Washington, Cuomo was asked if it's fair to say he's had to "walk a fine line" in his dealings with the president.
"Life is a fine line. Government is a fine line. Everything is a fine line. Just tell the truth," Cuomo said. "I said that to the president from day one. He has no problem telling me when he disagrees, and he tells me when he agrees. I have no problem telling him when I disagree and when I agree."
Trump, announcing the visit at Monday's coronavirus task force briefing, said he looked forward to meeting with the governor, after sparring with him for weeks over ventilators and accusing Cuomo of inflating New York's request for the critical machines.
He has said numerous times that Cuomo should've prepared better for the pandemic, even having aides pull research on how the governor denied ordering more ventilators for his state when given the chance a few years ago.
The New York governor, meanwhile, hasn't hesitated to likewise push back against the president and has at times criticized the federal government for not taking a stronger lead in its response.
Just last week, after the president lashed out at him on Twitter, Cuomo suggested the president should stop watching TV.
"First of all, if he's sitting home watching TV, maybe he should get up and go to work, right?" Cuomo said Friday. "Second, let's keep emotion and politics out of this, and personal ego if we can. Because this is about the people."
"You should not expect all these governors to run around and do an international supply chain while they are trying to put together their testing protocol in their state," Cuomo said Tuesday.
President Trump has also started to use Cuomo's own words to deflect criticism.
On Sunday, the president played a two-minute clip of Cuomo praising the federal government's response efforts at his own briefing earlier in the day.
“I just think it’s so good because it’s bipartisan," Trump said after playing the video, which failed to include any criticism from Cuomo. "I think it’s a tribute to New York. I think it’s a tribute to the federal government."
Trump partially misrepresented some of Cuomo's words again Monday night, claiming the governor agrees with him that testing should be entirely left to states.
"I want to draw your attention to Governor Cuomo's remarks during his press conference today. He said, ‘The president is right. The state's testing is up to the states to do,' which will implement the tests and logistically coordinate the tests," Trump said Monday, not mentioning what the governor had added right afterward -- that state labs need federal support with supply-chain problems.
Both the president and New York governor have taken opposite sides on GOP Maryland Gov. Larry Hogan's move to buy 500,000 tests from South Korea -- an action the president criticized and the governor pointedly praised Tuesday, saying he wished he'd thought of it first.
"Take a look at that map. The governor of Maryland could've called Mike Pence and could've saved a lot of money. Look at all those different places," Trump said Monday, pointing to a map of labs with access to testing in the Maryland area created by the White House.
"Could've saved a lot of money, but that's okay. I don't think he needed to go to South Korea. I think he needed to get a little knowledge. That would've been helpful."
Cuomo, meanwhile, in what could be seen as needling Trump by countering his criticism, called Hogan "a better governor" than he was for "thinking outside of the box."
"Why didn't I think of buying them from South Korea?" Cuomo said Tuesday morning. "It's not what states are usually in charge of… Let the federal government figure out national supply issues rather than have 50 states figure out how to be like Governor Hogan and by tests from South Korea."
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ABC News Ben Gittleson, Jordyn Phelps, Katherine Faulders and Elizabeth Thomas contributed to this report.