The source said the diplomatic mission would remain closed for at least a day. He did not say whether it will reopen on Thursday.
US officials in Yemen told media that a diplomatic vehicle had come under fire late Monday at a rebel-manned checkpoint close to the embassy.
A deteriorating security situation in the Yemeni capital has prompted the US State Department to deploy two warships to the Red Sea in case it needs to urgently evacuate embassy staff from the conflict-hit country in Southwest Asia.
The Russian Embassy in Sanaa was open on Wednesday with tightened security due to the ongoing hostilities, Timofei Boikov, a Russian diplomat in the country, told RIA Novosti.
"Security has been strengthened, but the consulate is open for business. Many other embassies are not providing consular services [at the moment]," Boikov said, adding there are about 2,000 Russian nationals in Yemen.
Boikov said there have been no requests for evacuation from the 451 Russians registered with the embassy.
Fierce fighting continued on Wednesday in and around Sanaa after Houthi rebels stormed the presidential residence in the capital overnight.
The Houthis, Yemen's main opposition movement, played a major part in ousting former President Ali Abdullah Saleh in 2012. Last September, the Houthi insurgency overran the capital and spread across western and central parts of the country.
