The planned visit comes amid elevated tensions between Russia and the NATO military alliance over Ukraine.
Lavrov and US Secretary of State Antony Blinken made no breakthrough in talks on the issue this week but agreed to keep talking to try to resolve a crisis that has stoked fears of a military conflict.
"The British foreign minister has asked for the opportunity to travel to Moscow for talks with Lavrov. Moscow has agreed for the visit in February," RIA quoted its source as saying.
Russia's foreign ministry did not immediately respond to a Reuters' request for comment.
The UK's foreign ministry declined comment.
A senior UK defence ministry source said on Saturday that Russian Defence Minister Sergei Shoigu had also agreed to talks in Moscow with his British counterpart Ben Wallace.
Shoigu had invited Wallace to Moscow on Friday, while Wallace had earlier this week extended a similar invitation for his Russian counterpart to visit London for talks.
"The Defence Secretary is glad that Russia has accepted the invitation to talk with his counterpart," the source said.
"Given the last defence bilateral between our two countries took place in London in 2013, Russian Defence Minister Sergei Shoigu has offered to meet in Moscow instead. The Secretary of State has been clear that he will explore all avenues to achieve stability and a resolution to the Ukraine crisis."
The British Ministry of Defence source said no timing for the meeting had been confirmed.
The UK and its NATO allies have expressed deep concern over Russia's military build-up on the border with Ukraine and have warned it of severe economic consequences if it invades.
Russia denies plans to invade Ukraine but is demanding legally binding security guarantees from the United States and NATO.
The UK said on Monday it had started supplying Ukraine, which is not a member of the NATO alliance, with anti-tank weapons to help it defend itself from a potential invasion.
