Mr Trump accused the National Security Agency (NSA) and FBI of giving out information illegally.
Some US media say top Trump aides were in constant communication with Russian officials during the election campaign.
Intelligence officials previously said they believed Russia tried to influence the vote in favour of Mr Trump.
Moscow has dismissed the claim as unsubstantiated.
Kremlin spokesman Dmitry Peskov said the latest reports of contacts with Trump aides were "not based on any facts".
On Thursday (local time) Rex Tillerson is due for his first meeting as US secretary of state with Russian Foreign Minister Sergei Lavrov, at a G20 gathering in the German city of Bonn.
The latest allegations look set to reignite tensions between the president and his own intelligence services, which flared during the 2016 election campaign.
They come a day after National Security Adviser Michael Flynn resigned amid a row over reports he discussed US sanctions by phone with a Russian diplomat before Mr Trump took office.
Leading Republicans have joined calls for a wide investigation into Mr Flynn's links with Russia, and the FBI and a Senate committee are also investigating.
Mr Trump appeared to attack the NSA and the FBI for what he described as leaks to the media about the contacts.
"Information is being illegally given to the failing @nytimes & @washingtonpost by the intelligence community (NSA and FBI?)," he tweeted.
But in another tweet he suggested that the reports were "non-sense" produced by his Democratic opponents.
"This Russian connection non-sense is merely an attempt to cover-up the many mistakes made in Hillary Clinton's losing campaign."
The tweets came after the New York Times quoted unnamed former and current officials as saying that the communications were intercepted as evidence of Russian hacking of the Democratic National Committee was coming to light.
Other media have since corroborated the reports.
The intercepted communications were said to be between Trump campaign officials and other associates on the one hand, and Russian intelligence and government officials on the other.
However, the officials interviewed by the Times said they had seen no evidence of the Trump team colluding in the hacking.
Mr Flynn and Mr Trump's former campaign manager, Paul Manafort, were said to be among those mentioned by the officials.
Mr Manafort told the newspaper: "I have never knowingly spoken to Russian intelligence officers, and I have never been involved with anything to do with the Russian government or the Putin administration or any other issues under investigation today.
"It's not like these people wear badges that say: 'I'm a Russian intelligence officer.'"
Mr Manafort and former Trump foreign policy aide Carter Page both left the campaign after reports emerged of their close links with Russia.
Photo: AFP Donald Trump has tweeted the reports are "non-sense" produced by his Democratic opponents.