Peskov dodged a question about Trump's news conference Thursday in which the president blamed the media for trying to thwart his plan to improve ties with Moscow.
Moscow believes that "Russia-U.S. ties are important for our two nations and the entire global community, so we believe that they deserve special attention," Peskov said in a conference call with reporters.
At his White House news conference, Trump denied that his campaign aides had been in touch with Russian officials before the U.S. election and said he had "nothing to do with Russia."
Asked whether the Kremlin watched the news conference, Peskov said Putin's administration was too busy with its domestic agenda.
Areas of possible cooperation or disagreements between the U.S. and Russia could only be determined after Putin and Trump have substantive talks, Peskov said, adding that it's not clear when that might happen.
"Only after they have a chance to have a detailed talk it would become clear where significant differences remain and where it's possible to find areas for cooperation," Peskov said.
He denied that Russian state-controlled television had been ordered to tone down fawning coverage of Trump, saying broadcasters don't take orders from the Kremlin.
A change of attitude, however, was clearly visible in Friday's news programs on Russian state TV, which gave little time to Trump compared with previous massive coverage of the U.S. president.
The shift in tone could reflect an attempt to dampen public expectations of a quick breakthrough in better relations with Washington that have been fueled by Trump's victory.
Members of the Kremlin-controlled parliament and pro-Kremlin media commentators, meanwhile, have increasingly voiced concerns about Trump's course.
The daily Komsomolskaya Pravda noted the difference between Trump's criticism of NATO during the campaign and his expressions of support for the alliance once in office and suggested a stiff drink might make it clearer.
"You need a bottle to figure out what the U.S. president's true position is," it said.
Valery Garbuzov, the head of the United States and Canada Institute, a government-funded think-tank, said in remarks carried by the Interfax news agency that Moscow and Washington haven't yet started specific discussions on issues such as the Ukrainian crisis and the Syrian war.
"In order to conduct a dialogue, it's necessary to rebuild mutual trust, which has been completely lost," he said.
Alexey Pushkov, the head of the information committee in the upper house of Russian parliament, tweeted: "It looks like Trump didn't expect such a powerful opposition to his decisions and plans."
"There is a high probability that it will not be Trump who will 'drain Washington's swamp,' but the swamp that will suck Trump in," Pushkov said. "And that will mark an end of his revolution that never began."