Asked by journalists about the Kremlin's links with the US, Peskov said: "I'd like to remind you of President Putin's words -- numerous times he has talked about his wish to build good, mutually beneficial relations based on mutual respect and equality."
"The President always said he expected a reciprocity from Washington," Peskov added. "Now we know that our bilateral relations are at the bottom so it's hard to make them worse, but we certainly hope for resuming a dialogue and we'll start a difficult and slow process of bringing the relations back to a constructive course."
Improving US ties with Russia was a key plank of US President-elect Donald Trump's foreign policy during the election campaign.
Many in Russia had expected Hillary Clinton -- who has been consistently critical of the Kremlin and is deeply unpopular in Moscow as a result -- to sweep to victory on November 8.
Following Trump's unexpected win, there are hopes of a fresh start -- though Trump told the New York Times this week that he was not looking to "reset" US-Russia relations.
Asked about Trump's comments, Peskov referred to Clinton's previous ill-fated attempt to "reset" relations while serving as Secretary of State.
"As for a reset, we can only agree with the President-elect because this word has embarrassed itself as the consequences of that reset are not the ones we'd like to see," Peskov said.
"The term doesn't matter. It's about a will and showing the readiness for normalizing the relations -- these are the most important things."
Relations between the US and Russia have deteriorated in recent years. They reached a low point in 2014, when Ukraine's pro-Kremlin leader was toppled in what Moscow describes as a Western-backed coup.
Russia's subsequent annexation of Crimea and support for rebels in eastern Ukraine prompted Western sanctions.
The West's policies on Ukraine, new US missile bases in Eastern Europe, and NATO troop deployments near Russia's borders are all viewed by the Kremlin as evidence of US aggression.
In an interview recorded earlier this year with the filmmaker Oliver Stone -- when the pollsters were predicting an overwhelming victory for Hillary Clinton in the US presidential election -- Putin vowed to respond.
"We need to take counter measures ... to pose a threat to rocket systems that are threatening to us," he told Stone in the director's latest controversial offering, "Ukraine On Fire."
In October, Russia's military sent nuclear-capable "Iskander" missiles to its Baltic enclave of Kaliningrad, within striking distance of Eastern European targets.
A senior Russian defense official said one of the missiles had been deliberately exposed to US spy satellites in what US intelligence officials interpreted as a "gesture to express disapproval" with NATO.
"Why are we reacting to NATO expansion so emotionally?" Putin asked in the Stone documentary, broadcast on Russian television on November 21, a national holiday in Ukraine which marks the toppling of the pro-Russian government.
"When a country becomes a NATO member, it becomes hard for it to resist the pressure from such a big country -- NATO's leader -- like the United States," he said.
The comments were interpreted as a shot across the bows of Clinton, who might have been on the cusp of entering the White House when the documentary aired.
But the geopolitical landscape has unexpectedly shifted in the meantime.
Trump, who won the US presidential election against the odds, campaigned on a platform of detente with Russia and has expressed reservations about NATO.
While the possibility of a dramatic fallout with the Kremlin remains real, so is the prospect of a warmer phase in relations between Moscow and Washington.