Trump, who has promised to end the war in Ukraine but not yet set out in public how he would do so, said last week the war was a bloodbath and his team had had "some very good talks".
In an interview aboard Air Force One on Friday, Trump told the New York Post that he had "better not say" when asked how many times he and Putin had spoken.
"He (Putin) wants to see people stop dying," Trump told the news outlet.
The White House did not respond to a request for comment outside normal business hours.
Kremlin spokesman Dmitry Peskov told the TASS state news agency "many different communications are emerging".
"These communications are conducted through different channels," Peskov said when asked by TASS to comment directly on the US report.
"I personally may not know something, be unaware of something. Therefore, in this case, I can neither confirm nor deny it."
The conflict in eastern Ukraine began in 2014 after a pro-Russian president was toppled in Ukraine's Maidan Revolution and Russia annexed Crimea, with Russian-backed separatist forces fighting Ukraine's armed forces.
Putin sent thousands of troops into Ukraine in 2022, calling it a "special military operation" to protect Russian speakers in Ukraine and counter what he said was a grave threat to Russia from potential Ukrainian membership of NATO.
Ukraine and its Western backers, led by the US, said the invasion was an imperial-style land grab and vowed to defeat Russian forces.
Moscow controls a chunk of Ukraine about the size of the American state of Virginia and is advancing at the fastest pace since the early days of the 2022 invasion.
Trump, author of the 1987 book Trump: the Art of the Deal, has repeatedly said he wants to end the war and that he will meet Putin to discuss it, although the date or venue for a summit is still not publicly known.
Saudi Arabia and the United Arab Emirates are seen by Russia as possible venues for a summit, Reuters reported in February.
On June 14, Putin set out his opening terms for an immediate end to the war: Ukraine must drop its NATO ambitions and withdraw its troops from the entirety of the territory of four Ukrainian regions claimed and mostly controlled by Russia.
The Kremlin has repeatedly urged caution over speculation about contacts with the Trump team over a possible peace deal.
Leonid Slutsky, head of the Russian parliament's international affairs committee, was cited by the state RIA news agency on Thursday as saying preparations for such a meeting were at "an advanced stage" and it could take place in February or March.
Putin last spoke to former US president Joe Biden in February 2022, shortly before Putin ordered thousands of troops into Ukraine.
On Friday, Trump said he would probably meet Ukrainian President Volodymyr Zelenskiy in the coming days to discuss ending the war.
Trump told the New York Post that he has "always had a good relationship with Putin" and that he has a concrete plan to end the war.
"I hope it's fast," Trump said.
"Every day people are dying. This war is so bad in Ukraine. I want to end this damn thing."
