Britain is considering allowing Russian officials to see Yulia Skripal over the Salisbury poisoning.
Ms Skripal spent three weeks in a critical condition after she and her father Sergei Skripal were exposed to Novichok on March 4, but her condition is now said to be improving rapidly.
The Russian Embassy said on Friday it "insists" on the right to see the 33-year-old.
A Foreign Office spokeswoman said: "We are considering requests for consular access in line with our obligations under international and domestic law, including the rights and wishes of Yulia Skripal."
Ms Skripal's father remains in a critical but stable condition in hospital.
It comes as the row between the Kremlin and Britain intensified.
On Friday, the Kremlin ordered Britain to reduce the number of diplomats in Moscow to the same amount that Russia has in London. Ambassadors from more than 20 countries were also called into the foreign ministry in Moscow to be told of the latest wave of retaliatory measures being imposed.
The Russian embassy has also accused British authorities of "another blatant provocation" after an Aeroflot flight was allegedly searched by Border Force officers at Heathrow.
An embassy official suggested that the jet, which had arrived from Moscow on a scheduled return trip, had been searched in connection with the diplomatic crisis over the Salisbury spy poisoning.
According to an embassy officer Border Force and customs officers at first tried to search the plane without the crew, although the captain was eventually allowed to be present.
"Today, we have witnessed another blatant provocation by the British authorities," the official said.
"Border Force and Customs officers have searched the aircraft that was conducting the Aeroflot flights 2582 / 2583, Moscow - London - Moscow. This kind of event is extraordinary."
The Metropolitan Police, which is coordinating the investigation into the poisoning, denied it was part of the search.
The Russian embassy said it had complained to British counterparts and called for an explanation.
The UK expelled 23 Russian diplomats in the wake of the poisoning of Mr Skripal, 66, and his daughter Yulia with the nerve agent Novichok.
Russia has told a number of countries - including Ireland, Germany, Italy, France, Spain, Finland, Poland, the Czech Republic, the Netherlands, Norway and Australia - they must send home the same number of diplomats as they had ordered to leave their nations.
So far, more than 150 Russian diplomats, some of whom are suspected spies, have been told to return home by the UK's allies.
On Friday afternoon, the Russian Embassy published a list of 27 questions it claimed it had for the British authorities over the treatment of the Skripals. The list included demands to know the exact nature of their condition and how they were being treated, and whether British doctors' approach "helped or harmed" the Skripals.
It also claimed Mr Skripal's niece had been inquiring after her uncle's health but that she had been ignored by the Foreign Office, and wondered why no images or footage of the Skripals alive in hospital had been published.
Scotland Yard believes Mr Skripal and his daughter, who was visiting him from Russia, first came into contact with the deadly chemical at his home. Detailed forensic testing revealed the highest concentration of Novichok was found on the front door.
The vast attempted murder investigation continues to unfold in the Wiltshire city.
Detective Sergeant Nick Bailey, who was also exposed to the nerve agent, was discharged from the hospital last week.