The assertion comes after Trump was repeatedly accused by the Clinton camp of having overly close ties to the Russian regime, connections that Trump consistently denied.
"During this entire period, we not only sent some signals through some representatives, or private messages," Russian Foreign Ministry spokeswoman Maria Zakharova said Thursday about the communications with the Trump staff.
"It was our clear position that we are ready for cooperation and working together and establishing normal relations," she added.
Russian Deputy Foreign Minister Sergei Ryabkov also said that officials had been in contact with members of Trump's entourage.
"I cannot say that all of them but quite a few have been staying in touch with Russian representatives," he told the state-run Interfax news agency.
Hope Hicks, a spokesperson for Trump's campaign, flatly denied the Russian statements, telling CNN, "This is not accurate."
The Clinton campaign did not respond to a request for comment.
A spokesman for the US State Department, however, said Thursday that it was not uncommon for foreign officials to have contact with US political campaigns.
"There are instances where other foreign governments ... have contacts with the different political parties during a campaign," Mark Toner told reporters at the State Department.
The Russian assertion comes after Democrats repeatedly accused Trump of being a puppet of Russian President Vladimir Putin during the course of the election season.
Former CIA Acting Director Michael Morell, for one, called Trump an "unwitting agent of the Russian Federation."
Morell, a Clinton backer, said the GOP candidate's positions on a variety of security issues hewed closely to the Kremlin's ideology.
Trump has said he wants to make improved ties with Moscow a central goal of his administration's foreign policy. He repeatedly cast doubt on Russia's involvement in cyberattacks against the Democratic National Committee, despite the Director of National Intelligence and Department of Homeland Security issuing a joint statement saying the Russian government was to blame.
A Russian official told CNN Thursday that that the Russian Foreign Ministry had offered to establish contact with both the Clinton and Trump teams in the run up to the US election. It's a move which the Ministry says is normal diplomatic practice "to maintain relations with the two obvious candidates."
The official said members of Clinton's team visited Moscow unofficially and held meetings with high-level representatives from both the Kremlin and the Foreign Ministry.