Sergei Skripal and his daughter Yulia were poisoned with the nerve agent in Salisbury in March, 2018.
Mr Skripal's niece Viktoria Skripal has confirmed to The Daily Telegraph she received a phone call from Yulia last month with an update on her uncle's condition.
Russian newspaper Moskovsky Komsomolets published a transcript of the two women's conversation, in which Yulia told her cousin she had fully recovered from the poisoning but her father was "in a tight spot".
The transcript showed the poisoned pair live in separate flats due to the pandemic, with Ms Skripal saying of her father: "He feels fine but I don't see him because of the lockdown."
Mr Skripal had a tracheotomy following the attack and now uses a tracheal tube to breathe, she added.
Viktoria confirmed the transcript's veracity to the Telegraph.
The Skripals survived the attack but the incident later claimed the life of Dawn Sturgess after she came into contact with a perfume bottle believed to have been used in the attack before being discarded.
Her partner, Charlie Rowley, was left seriously ill but recovered.
The flat where Ms Sturgess was fatally poisoned was demolished in late October.
Two Russian nationals have been accused of travelling to the UK to try to murder Mr Skripal with Novichok.
The suspects - known by aliases Alexander Petrov and Ruslan Boshirov - were seen on CCTV footage in Salisbury the day before the attack.
Moscow has repeatedly denied any involvement, with President Vladimir Putin claiming the two suspects were civilians.
Reporting by the Press Association
