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".