Britain backed sanctions against Russia in response to its annexation of Crimea in 2014, and the countries have clashed on other issues, including Russia's support for Syrian President Bashar al-Assad in the civil war in his country.
However, Mr Johnson has accepted the invitation from counterpart Sergei Lavrov and the British government said that it was "clear that the UK will engage with Russia where it is in our national interest to do so".
"Discussions will focus on the UK-Russia relationship and current international issues including Syria and Ukraine, where we continue to have significant differences," a Foreign and Commonwealth spokeswoman said.
"This is not a return to business as usual and the Foreign Secretary will continue to be robust on those issues where we differ."
The Kremlin calls the hacking allegations "baseless".
The last UK foreign minister to visit Moscow was William Hague in 2012, the Foreign Office said, adding that details of the exact timing of Mr Johnson's trip would be confirmed in due course.