Moscow has dismissed the claim, describing it as based on unsubstantiated facts.
Kremlin spokesman Dmitry Peskov said on Wednesday that the latest reports of contacts with Trump aides were "not based on any facts".
On Thursday 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, says BBC Washington correspondent Gary O'Donoghue.
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 on Wednesday Adam Schiff, a Democrat on the House of Representatives Intelligence Committee, said he would push for the committee's investigations into Russian hacking to be expanded to include Mr Flynn's phone calls, Reuters news agency reported.
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