The visit could signal an improvement in relations after years of antagonism. It comes after Johnson in April cancelled a planned trip at the last minute over Russia's support for the Syrian regime of President Bashar al-Assad.
Britain's outspoken foreign secretary himself said he holds out little hope that ties with Moscow could undergo a full-blown transformation.
In an interview with Polish news agency PAP ahead of his Russia visit, Johnson said he was "no cold warrior", but he did "not believe for a second that relations with Russia can be reset."
The Russian foreign ministry's spokeswoman Maria Zakharova said on Thursday the diplomatic chiefs planned to "look for ways to normalise and activate the bilateral relationship."
"Unfortunately, cutting short bilateral dialogue with Russia was London's choice," Zakharova said, adding that this was "completely unfounded."
She called the visit "long-awaited."
Relations between London and Moscow soured after Britain sought to prosecute suspects in the killing of Kremlin critic and former spy Alexander Litvinenko, murdered by radiation poisoning in London in 2006.
Britain has also been a fervent supporter of Western sanctions against Russia over its role in the Ukraine conflict and annexation of Crimea in 2014.
The relationship suffered further blows after Russia's intervention in the Syria conflict on the side of the Damascus regime in September 2015.
'What's the Syria plan'?
Johnson, who in his capacity as mayor of London called Russian strongman Vladimir Putin a "ruthless and manipulative tyrant" in 2015, is expected to make Syria a key focus of the talks with Lavrov.
Johnson told the PAP agency that Britain and Russia still "have huge differences on Syria."
"I would be interested in how the Russians see the future of Syria: Assad is still on the throne but what's the plan? He's not a very popular person, so how is it going to function?" Johnson asked.
Speaking at the United Nations in September, the top British diplomat denounced Russia-led Syria peace talks and said Britain would only support a solution that includes "a transition away from Assad."
'Weaponing information'
Concerns are also running high in Europe over the Kremlin's use of cyber-tactics and misinformation to cause political destabilisation around the continent and to draw former Soviet states back into its embrace.
Following the 2016 referendum on Brexit, in which Johnson campaigned to leave the EU, Britain has joined the growing number of Western countries accusing Russia of interfering in their political systems.
British media regulators have taken actions against Kremlin-funded news outlets including Russian television RT UK, which provoked anger in Moscow.
In a major foreign policy speech last month, British Prime Minister Theresa May accused Russia of "weaponing information" and "threatening the international world order on which we all depend."
"I have a very simple message for Russia. We know what you are doing. And you will not succeed," May said at the time, a phrase that was greeted with mockery in Moscow.
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