President Vladimir Putin believes Donald Trump is satisfied with his assertions Russia did not meddle in the US presidential election.
Speaking at the end of a G20 summit in Germany, where the two leaders met face-to-face for the first time, Mr Putin said he believed he had been able to establish a personal relationship with Mr Trump.
The initial groundwork had now been laid for an improvement in US-Russian ties, he said.
Their meeting on the sidelines of the Hamburg summit was subject to intense scrutiny following allegations that Moscow had tried to help Mr Trump win the White House, and a Washington investigation into the Russia ties of Trump associates.
Mr Putin was pressed by reporters at a news conference to share details of the discussion he had with Mr Trump about alleged Russian election interference.
Russia has denied trying to influence the US election.
"(Mr Trump) asked a lot of questions on this subject. I, in as much as I was able, answered these questions. It seems to me that he took these (answers) on board and agreed with them, but in actual fact, it's best to ask him how he views this," Mr Putin told one reporter.
Pressed again later in the same news conference by reporters about what precisely Mr Trump had told him, Mr Putin said: "He started to ask pointed questions, he was really interested in particular details. I, as much as I could, answered him in a fairly detailed way.
"I believe it would not be entirely appropriate on my part to disclose details of my discussion with Mr Trump. He asked, I answered him. He asked pointed questions, I answered them. It seemed to me that he was satisfied with those answers," Mr Putin said.
Mr Putin also spoke warmly of Mr Trump's personal qualities.
The two men have spoken by telephone since Mr Trump won the US presidential election last year, but had not met until the summit in Hamburg.
"I believe that we have established personal relations already," Mr Putin told a news conference.
"The TV Trump is very different from the person in reality. He is absolutely precise, he reacts as you would expect to his interlocutor, he analyses fairly quickly, answers questions that are put to him.
"It seems to me that if we build our relations the way that our conversation went yesterday, then we all have grounds to believe that we can, at least in part, restore the level of cooperation that we need," Mr Putin said.
He said in particular that the Trump administration was taking a more pragmatic stance on the conflict on Syria.
A ceasefire deal for southern Syria that was announced during the summit was the result of that new approach, Mr Putin said, and represented a "breakthrough".
But while the leaders of Russia and the United States appeared to have a successful meeting, a gulf between Mr Trump and many of the European leaders remains.
French President Emmanuel Macron says he has not given up on trying to get Mr Trump to change his mind about withdrawing the United States from the Paris climate change agreement.
The final statement from the Group of 20 leaders meeting in Hamburg on Saturday exposed the divide that remains between the United States and other G20 members on the 2015 Paris accord aimed at combating climate change.
But Mr Macron also said he would host a summit on December 12 to move the Paris deal forward.
He said the summit would aim to mobilise private and public financing for the projects committed to under the Paris agreement.