At a news conference on Wednesday, Trump said he believes Russia's intelligence agencies were responsible for hacking computers belonging to the Democratic Party. Russia has consistently denied any involvement in the US election.
However, in a conference call on Thursday, the Kremlin urged closer ties with the US.
"President Putin with his actions and his statements has clearly and undoubtedly showed his readiness to respect our partners, but this respect can only be mutual, it cannot be unilateral," Russian President Vladimir Putin's spokesman Dmitry Peskov told reporters.
Peskov expressed hope that Putin and Trump would get along.
"There should be more mutual respect in Russia-US ties, from our viewpoint, much more," he said.
"Indeed, Moscow hopes that our presidents will get along. At least, Mr. Trump was talking about his readiness to have a dialogue," he added.
"It does not mean a readiness to agree with each other on everything. That's what Moscow does not expect, but a dialogue is what we should hope for in search of many complicated situations."
Peskov also responded to remarks this week from prospective Secretary of State Rex Tillerson who claimed Russia's annexation of Crimea in 2014 was an "illegal action" and that Moscow "must be held to account for its actions."
"This is his position, we will take it into consideration," Peskov added.
"Of course, Russia will patiently keep explaining the matter [to him]. Of course, we don't agree with [Tillerson's] wording and will be reasonably explaining our position [to him]."
The comments from Russia come a day after Peskov rejected reports that Trump was the subject of "Kompromat" -- a Russian term for compromising information intended to be used against someone.
"Clearly there are those who are creating hysteria, who are trying to support this witch hunt, and President-elect Trump himself described it like this," Peskov said.
CNN reported Tuesday that intelligence officials last week presented President Obama and Trump with a two-page synopsis of memos about the President-elect.
Multiple US officials with direct knowledge of the briefings told CNN that the synopsis included allegations that Russian operatives claim to have compromising personal and financial information about Trump.
It also contained allegations of a continuing exchange of information during the campaign between Trump surrogates and intermediaries for the Russian government.