The United States has consulates in three Russian cities: St. Petersburg, Ekaterinburg and the far eastern port, Vladivostok.
Tillerson and Lavrov also agreed to a meeting in September, according to the official.
In July, Moscow ordered the United States to slash its diplomatic and technical staff in Russian Federation by more than 50 per cent, expelling 755 people, in response to fresh U.S. sanctions over interference in the 2016 presidential election.
The spokesperson of the U.S. Department of State, Heather Nauert, said in a statement on Thursday that the move was to achieve parity in diplomatic missions.
The score-settling that started in December is the broadest involving the two countries' foreign outposts since 1986, when the United States and the Soviet Union expelled dozens of each other's diplomats.
Meanwhile, Russian consul in San Francisco Sergei Petrov told RIA-Novosti news agency that his consulate will act in accordance with the decision that will be taken by the Russian leadership regarding the United States decision.
New Russian Ambassador Anatoly Antonov, who arrived in Washington only hours after the USA announcement, cited a maxim of former Soviet leader Vladimir Lenin as he urged caution and professionalism.
Russian news agencies quoted him as saying: "We have to act calmly and professionally". USA officials pointed out that Russian Federation, when it ordered the cut in US diplomats, had argued it was merely bringing the size of the two countries' diplomatic presences into "parity". The two are expected to meet at the UN General Assembly in NY next month.
"But this step also highlights once again the disconnect between Trump and his administration - for the last time we heard from the President on this issue he joked that he thanked Putin for reducing the size of the USA government", he added.
"... Another unfriendly step of the US authorities, which, first of all, will hit hard on Russian citizens residing in the consular district, as well as on American citizens".
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Her auto had stalled in floodwaters and when she grabbed her daughter and got out of the vehicle , floodwater swept them away. The rescuers detached the child from her mother's body and tried to revive the mother, but she did not respond.
Russian Foreign Ministry spokeswoman Maria Zakharova claims U.S. The State Department says it will "secure and maintain the properties in keeping with our responsibilities".
Mr Lavrov hinted on Friday Russia might look at ordering further reductions in U.S. embassy staff, suggesting Moscow had been generous last time by allowing Washington to keep "more than 150" extra people.
"It is an important and needed response, but it is not proportionate", said Michael McFaul, who served as ambassador to Moscow during President Barack Obama's administration. It is still unclear how many Russian employees will be affected.
The State Department isn't commenting specifically on whether officials plan to search the premises.
Putin ordered 755 American diplomatic personnel be cut and two diplomatic facilities in Moscow be shut down. Still, Nauert noted in her statement that the USA has "fully implemented" the Russian order, which had set its deadline for Friday.
Russian Foreign Minister Sergei Lavrov says that Moscow is examining the US demand for the closure of three diplomatic missions in the United States and will respond after further study.
The reductions are having consequences for Russian Federation. "They are forcing us into it", Zakharova said. Russians would no longer be able to apply for visas at United States consulates in St. Petersburg, Yekaterinburg and Vladivostok. -Russian cooperation that have transcended the worsening ties.
Earlier this month, President Trump was asked about Putin's retaliation. The U.S. says the truce has largely held.
Information for this article was contributed by Mark Landler of The New York Times; by Josh Lederman, Vladimir Isachenkov, Jim Heintz and Garance Burke of The Associated Press; and by Anne Gearan of The Washington Post.