"The US president has expressively voiced his willingness to re-energise dialogue … [Putin] said long ago he wanted to normalise our ties … but stressed it was not mutual. It was the first time that the US president had reciprocated", he said on a Rossiya 1 television show.
Peskov said the heads of state spoke confidently about the need to maintain contacts.
"Putin and Trump spoke quite confidently about the need to continue their contacts at the highest level. At least they said they were not parting for long", Peskov added.
The two presidents met in private on 28 June and talked for one and a half hours on the sidelines of the G20 Summit in Japan. Both presidents have described the talks as positive, with Trump calling the negotiations a "great discussion".
During their much-anticipated meeting, Putin and Trump discussed the the situation in Venezuela, Syria, Ukraine, strategic stability and arms control, as well as alleged Russian meddling in the US electoral process, according to the Kremlin. The Russian president also invited his US counterpart to come to Moscow to attend next year's celebration of the 75th anniversary of the allied victory in World War II.
At the talks, the sides agreed to begin a dialogue on the New START nuclear arms control treaty. The fate of the treaty, which is due to expire in February 2021, has been unclear due to a decision announced by the US president to suspend obligations under the 1987 Intermediate-Range Nuclear Forces (INF) Treaty.
In February, the United States formally suspended its obligations under the nuclear disarmament accord and triggered the six-month withdrawal process, making 2 August the deadline. Moscow responded by suspending its participation in the treaty as well.