"Trump has not taken office yet. The United States has the incumbent president and the Russian side works with him," Peskov said, TASS reported.
He pointed out he had never mentioned any contacts between Trump's team and the Kremlin.
"I did say, though, that both experts, political scientists and staff of Russia's offices abroad maintained contacts with the whole spectrum of the political elite on the regular basis. That's standard practice," he said, adding he had never heard Trump had visited Moscow after his nomination as a presidential candidate.
Read alsoThe Telegraph: Trump-Putin alliance sparks diplomatic crisis as British ministers demand assurances from U.S. over RussiaPeskov said Trump's team "is still being formed" and it was "too early to say anything" about expectations regarding a future U.S. Administration. Peskov warned against trying to anticipate events.
"Judgement should be made on the basis of certain steps and decisions by a future Administration," he said.
Peskov stressed that the Kremlin had no contacts with anyone in the U.S. president-elect's entourage. During his trip to the United States Peskov had no meetings with any politicians close to Trump.
About the message from Russian President Vladimir Putin, which Trump described as "beautiful," Peskov said "it was a message of greeting sent on the day the election returns were made public."
"In this message Putin congratulated Trump upon his election victory," Peskov said. Its brief summary was published on the Kremlin's website.
Any contacts and visits may be discussed only after the official inauguration, the Kremlin spokesman said. Asked about his earlier statements that Putin-Trump contacts might be possible before the latter's inauguration in January 2017 Peskov said contacts did not necessarily imply a personal meeting.
"There may be a written message, a telephone conversation or a personal meeting," he said. "It cannot be ruled out."
