Rubio teases details of potential Trump, Putin in-person meeting after Russia-Ukraine ceasefire talks stall

Meanwhile, Rubio - who attended Pope Leo XIV's inaugural mass in Rome on Sunday - said the Vatican has offered to host a direct meeting between Ukraine, Russia and possibly other parties. 

"Obviously, the Vatican has made a very generous offer to host anything - by the way, not just a meeting between Zelenskyy and Putin, but any meeting, including at a technical level, you know - any meetings that need to be hosted, they've expressed a willingness to do so. So it's a very generous offer that may be taken up on," Rubio told CBS' "Face the Nation" in an interview that was recorded on Saturday. "I mean, it would be a site that all parties would feel comfortable. So hopefully we'll get to that stage where talks are happening on a regular basis, and that the Vatican will have the opportunity to be one of the options." 

CBS host Margaret Brennan asked Rubio if he spoke with Lavrov about lining up a face-to-face meeting between Trump and Putin. 

"Well, we talked about a variety of things," said Rubio. "I wanted to get his readout on his view of how the talks went yesterday. They were not a complete waste of time. For example, there were 1,000 prisoners that are going to be exchanged, and that, from a humanitarian standpoint, is very positive. He explained to me that they are going to be preparing a document outlining their requirements for a ceasefire that would then lead to broader negotiations."

Rubio said the Ukrainians will be working on their own proposal coming soon, and he hoped proposals from both sides would be "serious and viable."

"So we'll have to wait and see. But he wanted me to know, and he communicated in our call, that their side will be working on a series of ideas and requirements that they would have in order to move forward with a ceasefire and further negotiations," he said.

Rubio said the U.S. is "testing" whether the Russian are just "tapping" them along, as Trump has suggested could be the case. 

"If, on the other hand, what we see is not very productive, perhaps we'll have a different assessment. I also agree that, ultimately, one of the things that could help break this logjam - perhaps the only thing that can - is a direct conversation between President Trump and Vladimir Putin. And he's already openly expressed a desire and a belief that that needs to happen, and hopefully that'll be worked out soon as well," he added.

Pressed on whether the in-person talks between Trump and Putin were being planned, Rubio reiterated that the president had already made that offer publicly. 

"The mechanics of setting that kind of meeting up would require a little bit of work, so I can't say that's being planned as we speak in terms of picking a site and a date," Rubio said. "But the president wants to do it. He wants to do it as soon as feasible. I think the Russian side has also expressed a willingness to do it. And so, now it's just a question of bringing them, bringing everyone together, and figuring out where and when and that meeting will happen and what it will be about."

Vance and Rubio later met with Italian Prime Minister Georgia Meloni and European Commission President Ursula von der Leyen for discussions on trade, the war in Ukraine and NATO spending, according to a spokesperson for the vice president. 

"The individual countries within Europe are important allies of the United States. But, of course, we have some disagreements, as friends sometimes do, on issues like trade, and we also have many agreements and many things we can work on together, and I'm looking forward to the conversation," Vance told reporters at the top of the meeting. 

After the meeting, the vice president's office released a statement saying that "the leaders discussed their shared goal of ending the bloodshed in Ukraine and provided updates on the current state of negotiations for a ceasefire and lasting peace." 

Fox News' Meghan Tomes and The Associated Press contributed to this report.