Alexei Navalny's mother has been told to agree to a "secret" burial for the Putin critic within three hours, Mr Navalny's spokeswoman says.
Otherwise, she was told, he would be buried at the Arctic Circle penal colony where he died a week ago.
Mr Navalny's mother has said she has been forced to sign a death certificate saying he died of natural causes.
But Mr Navalny's widow, Yulia, has said he was killed on the orders of Russian President Vladimir Putin.
The Kremlin has denied the allegations, calling Western reaction to the death "hysterical".
Mr Navalny died on 16 February. Prison officials said he had fallen ill following a "walk".
Mr Navalny's team has offered security officers €20,000 ($22,000; £17,000) in reward and assistance in leaving Russia in exchange for information about the Russian opposition leader's death in prison.
Mr Navalny's spokeswoman said the opposition leader's mother, Lyudmila, had refused to negotiate with the authorities as they "are not authorised to decide how and where her son should be buried".
She is demanding compliance with the law, according to which investigators are obliged to hand over the body within two days of establishing the cause of death," Kira Yarmysh said.
These two days are due to expire on Saturday.
On Thursday US President Joe Biden met Mr Navalny's widow and his daughter in San Francisco.
"The president expressed his admiration for Alexei Navalny's extraordinary courage and his legacy of fighting against corruption and for a free and democratic Russia," the White House said in a statement.
A day later, the US announced more than 500 new sanctions against Russia over its invasion of Ukraine and Mr Navalny's death.
These include measures against Russia's main card payment system, financial and military institutions, and officials involved in Navalny's imprisonment.
US targets Russia with more than 500 new sanctions
The EU has also announced new sanctions on access to military technology.
For years Mr Navalny was most high-profile critic of the Russian leader.
In August 2020, the former opposition leader was poisoned using the Novichok nerve agent by a team of would-be assassins from the Russian secret services.
Airlifted to Germany, he recovered there before returning to Russia in January 2021, where he was imprisoned.
Attempts at commemorating his death have been met by a heavy-handed response from Russian authorities, with makeshift monuments cleared and hundreds arrested.
