Russia has been pressing demands that the US give it access to two diplomatic compounds seized in the US last year.
After high-level talks between both sides, one Russian official involved said the row had "almost" been resolved.
Russia has been angered by the move, with Foreign Minister Sergei Lavrov calling it "daylight robbery".
In December the US expelled 35 Russian diplomats and shut the compounds over suspicions of meddling in US elections.
The talks saw US Undersecretary of State Thomas Shannon host Russian Deputy Foreign Minister Sergei Ryabkov in Washington yesterday.
Ryabkov sounded positive after three hours of talks with the American diplomat.