The Russian Embassy in London on Saturday requested a meeting between its ambassador and Foreign Secretary Boris Johnson over the Salisbury poisoning.
A Russian Embassy spokesperson said it was "high time" for a meeting to discuss the investigation as well as a "whole range of bilateral issues", the BBC reported.
Current interaction between the Russian Embassy and the Foreign Office was "utterly unsatisfactory", the official said. However, the British Foreign Office said it was Russia's response that had been "unsatisfactory".
Britain had blamed Moscow for the nerve agent attack on former Russian double agent Sergei Skripal and his daughter Yulia in Salisbury on March 4. But Moscow denied it.
A British 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 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," it said.
On the other hand, a Russian Embassy spokesperson was quoted by TASS as saying: "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 come up with a constructive response and such a meeting will be organized in the near future".
The British Foreign Office said it would respond to the meeting request in due course.
The request followed criticism from the Russian Embassy after the British government's refusal to grant a visa to Yulia Skripal's cousin Viktoria 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".
Salisbury District Hospital has said Sergei Skripal was responding well to treatment and "improving rapidly" while his daughter was conscious and talking in hospital.
So far, more than 20 countries have expelled Russian envoys in solidarity with the UK over the crisis. Russia's request for a new joint investigation was voted down at the international chemical weapons watchdog at The Hague on April 4.
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".
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