President Donald Trump has urged Russia to stop its attacks in Ukraine and suggested Ukrainian President Volodymyr Zelensky was ready to give up Crimea as the price of a peace deal with Russia.

Speaking to reporters in New Jersey, Trump said he was disappointed that Russia has continued to attack Ukraine, and said his one-on-one meeting with Zelensky at the Vatican on Saturday had gone well.

Asked if Zelensky might be ready to give up Crimea – the Black Sea peninsula seized by Russia in 2014 – as part of a future peace deal with Moscow, Trump said: "Oh, I think so, yeah. Look, Crimea was 12 years ago."

Trump then blamed his Democrat predecessors, Presidents Barack Obama and President Joe Biden, for having allowed Russia to take Crimea "without a shot being fired".

"So, don't talk to me about Crimea. Talk to Obama and Biden about Crimea. And remember, this is Biden's war. This isn't Trump's war. I came in to try and solve a problem. And the problem is so many people are being killed."

US proposals on ending the three-year war in Ukraine have called for Washington's recognition of Moscow's control over Crimea as well as de facto recognition of Russia's hold on other parts of Ukraine.

In contrast, European and Ukrainian counter-proposals defer detailed discussion about territory until after a ceasefire is concluded.

Ukrainian President Volodymyr Zelensky says Ukraine is still fighting in Kursk. – Reuters

On Zelensky, Trump said: "I see him as calmer. I think he understands the picture, and I think he wants to make a deal."

US Secretary of State Marco Rubio, meanwhile, said that the Trump administration might abandon its attempts to broker a deal if Russia and Ukraine do not make headway.

"It needs to happen soon," Rubio told the NBC program Meet the Press. 

"We cannot continue to dedicate time and resources to this effort if it's not going to come to fruition."

Trump and Zelensky, in Rome for the funeral of Pope Francis, met in a Vatican basilica on Saturday to try to revive faltering efforts to end the war in Ukraine.

The meeting was the first between the two leaders since an angry encounter in the White House Oval Office in February and comes at a critical time in negotiations aimed at bringing an end to the conflict.

Trump rebuked Russian President Vladimir Putin after that meeting, saying on social media that there is "no reason" for Russia to shoot missiles into civilian areas.

In a pre-taped interview that aired on the CBS program Face the Nation on Sunday, Russian Foreign Minister Sergei Lavrov said Russia would continue to target sites used by Ukraine's military.

When asked about a Russian strike on Kyiv last week that killed civilians, Lavrov said that "the target attacked was not something absolutely civilian" and that Russia targets only "sites which are used by the military".

Zelenskiy wrote on the messaging app Telegram that his top military commander reported that Russia had already conducted nearly 70 attacks on Sunday.

"The situation at the front and the real activity of the Russian army prove that there is currently insufficient pressure on Russia from the world to end this war," Zelensky said.

US Security Adviser Mike Waltz says President Donald Trump is 'determined to use both carrots and sticks' in Ukraine-Russia negotiations. – Fox News via Reuters

Ukrainian and European officials have pushed back against the US proposals on how to end the war.

German Defence Minister Boris Pistorius said that Ukraine should not agree to the American proposal, saying it went too far in ceding swathes of territory in return for a ceasefire.

Mike Waltz, Trump's national security adviser, said the US president had "expressed his frustration" at both Putin and Zelensky but remained determined to help negotiate an agreement.

Waltz also said the United States and Ukraine would eventually reach an agreement over rare earth minerals.

Chuck Schumer, the top US Senate Democrat, said he was concerned Trump would "cave in to Putin".

"To just abandon Ukraine, after all the sacrifice that they made, after so much loss of life, and with the rallying of the whole West against Putin, it would just be a moral tragedy," Schumer said on CNN's State of the Union program.