The Russian Embassy in London has requested a meeting between its ambassador and Foreign Secretary Boris Johnson over the Salisbury poisoning.
An embassy spokesperson said it was “high time” for a meeting to discuss the investigation, as well as a “whole range of bilateral issues”.
Current interaction between the Russian Embassy and the Foreign Office was “utterly unsatisfactory”, they said.
The Foreign Office said it was Russia’s response that had been unsatisfactory.
A Foreign Office statement confirmed it had received the meeting request and said: “It’s over three weeks since we asked Russia to engage constructively and answer a number of questions relating to the attempted assassinations of Mr Skripal and his daughter.
“Now, after failing in their attempts in the UN and international chemical weapons watchdog this week and with the victims’ condition improving, they seem to be pursuing a different diversionary tactic.”
Sergei and Yulia Skripal were poisoned with a toxic nerve agent called Novichok in Salisbury more than a month ago.
The UK government claims Russia is behind the attack but Moscow has denied all involvement.
In a statement sent to the BBC an embassy spokesman said: “We believe that it is high time to arrange a meeting between Ambassador Alexander Yakovenko and Foreign Secretary Boris Johnson.”
It added that the ambassador had already sent a note to the foreign secretary, and it hoped “the British side will engage constructively and that such meeting is arranged shortly”.
The Foreign Office said it would respond to the meeting request in due course.
The request follows criticism from the Russian Embassy after the British government’s refusal to grant a visa to Ms Skripal’s cousin, Viktoria Skripal, to visit the UK.
On Friday the Home Office said the application did not comply with immigration rules.
But the Russian Embassy said Sergei and Yulia “remain hidden from the public”.
“The stubborn refusal to cooperate, to provide transparency and to answer the numerous questions means Britain has something to hide,” an embassy spokesman said.
Viktoria Skripal later told the BBC she did not have enough money in her bank account to satisfy the visa requirements.
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Mr Skripal was jailed by Russia for spying for Britain’s intelligence service, but released as part of a spy swap between the US and Russia in 2010.
His daughter Yulia was visiting him in the UK when the attack happened on 4 March.
Salisbury District Hospital has said Mr Skripal, is responding well to treatment and “improving rapidly”.
His daughter Yulia is conscious and talking in hospital.
A diplomatic crisis between Russia and the West has followed, with more than 20 countries expelling Russian envoys in solidarity with the UK.
Russia’s request for a new, joint investigation was voted down at the international chemical weapons watchdog at The Hague on 4 April.
Two days later, at a UN Security Council meeting, Moscow’s UN ambassador Vasily Nebenzia said Britain’s main goal had been “to discredit and even delegitimise” Russia with “unsubstantiated accusations”.
But Britain’s UN representative Karen Pierce said the UK’s actions “stand up to any scrutiny”.