Boris Johnson will warn Russia to stop cyber attacks which threaten Britain's national security or face retaliation of a similar kind from the UK.
He will say the UK has no malign intentions online but has the technical capability to fight cyber espionage.
Mr Johnson is the first UK foreign secretary to visit Russia in five years.
His trip follows Theresa May's accusation last month that Russia was trying to "undermine free societies".
BBC diplomatic correspondent James Robbins says that relations between the two countries are "at best very bad, at worst, appalling".
The UK prime minister warned in November about the risks of Russia's "sustained campaign of cyber espionage and disruption".
Her criticisms were repeated by Ciaran Martin, chief executive of GCHQ's National Cyber Security Centre, who said that Russia was "seeking to undermine the international system".
At a meeting in Moscow later, Mr Johnson will tell his counterpart, Russian Foreign Minister Sergei Lavrov, that the UK does not accept Russia's "hostile" behaviour.
Ahead of that trip, Mr Johnson said that relations between the UK and Russia "haven't been so bad for a very long time".
"There are areas in which Russia is behaving in a more hostile way towards our interests than at any time since the end of the Cold War.
"I will make clear... that there are things that we find extremely difficult to accept, and we can't accept."
The two politicians are also set to discuss the threat posed to global security by North Korea, the search for a political settlement in Syria and preservation of the Iranian nuclear deal.
Mr Johnson will raise the subject of the safety of English football fans when Russia hosts the World Cup next year.
Russians were blamed for injuring more than 100 England supporters in Marseille during the 2016 European Football Championships.
'Not business as usual'
So what's the point of this meeting, which has been cancelled twice before?
From Britain's perspective, it's to persuade Russia - as a fellow permanent member of the UN Security Council - to co-operate against common threats: from North Korea in particular, but also from a ruined Syria where no peace settlement is in sight.
The Russian Foreign Ministry says Russia is interested in "seeking a way to normalise relations with Britain, and to reactivate cooperation".
But Boris Johnson says it cannot simply be "a return to business as usual". There are low expectations of any substantial progress.
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