"There is indeed such a possibility. The details will be agreed upon," the source said when asked whether the visit could take place within the next couple of weeks.
No specific date has been set.
The report comes as U.S. officials continue diplomatic efforts aimed at ending Russia's war against Ukraine.
Kremlin spokesperson Dmitry Peskov said on Oct. 9 that Moscow expected Witkoff and Kushner could return but did not know when the visit would take place.
"Their visit can be expected. But we don't know whether it will take place within that timeframe," Peskov said when asked whether the U.S. envoys could travel to Moscow in October.
He added that Russian President Vladimir Putin was "always glad" to meet them.
Peskov said Moscow would be prepared to receive Witkoff and Kushner if they wanted to present ideas discussed with Ukrainian and European officials during talks in Miami.
Axios previously reported that the two U.S. envoys could travel to Moscow later in October for further talks with Putin after meeting Ukrainian negotiators and European officials.
Witkoff and Kushner held talks with Putin in Moscow on Sept. 5 that lasted more than three hours.
The U.S. envoys presented ideas for a possible settlement of Russia's war against Ukraine, but the meeting produced no public breakthrough.
The two later traveled to Kyiv for talks with President Volodymyr Zelenskyy and Ukrainian and European officials.
The latest reports of a possible Moscow visit also follow Trump's Oct. 9 phone call with Putin, during which the two leaders discussed the war and reached an agreement on additional Russian diesel supplies to U.S. and global markets.