Navalny, the best-known political opponent of President Vladimir Putin, died suddenly a week ago in the Arctic penal colony where he had been serving sentences totalling more than 30 years.
His mother Lyudmila, 69, has been demanding for days that authorities hand over his body to be buried in a way that will allow his friends, family and supporters to pay their respects.
Navalny's spokeswoman Kira Yarmysh posted on X: "An hour ago, an investigator called Alexei's mother and gave her an ultimatum. Either she agrees within three hours to a secret funeral without a public farewell, or Alexei will be buried in the penal colony."
Yarmash said Navalny's mother was refusing and continuing to demand that his body be handed to her.
There was no immediate comment from the authorities.
Navalny's family and supporters have accused Putin of having him killed, an accusation that the Kremlin has angrily rejected.
Supportes of Alexei Navalny have accused the Russian president of being behind his death. (AP PHOTO)
He had survived a poisoning attempt in 2020 and years of harsh treatment in prison, including long spells in solitary confinement.
Putin makes a point of never mentioning Navalny's name, and has not commented on the matter in public since his death.
The Kremlin has said it had no hand in his death and that the circumstances are being investigated.
The stand-off over the release of his body appears to reflect Kremlin concern that a big public funeral for Putin's leading critic could become a focal point for unrest, weeks before an election in which the Kremlin leader is seeking six more years in power.
He faces three other candidates, none of them critical of Putin.
Rights group OVD-Info says 400 people were detained in the days following Navalny's death for trying to commemorate it in public.
In a 95-second video released on Thursday, Lyudmila Navalnaya said authorities were trying to "blackmail" and threaten her into agreeing to a non-public burial, something she said she would not accept.
"They want this (burial) to be done secretly, with no farewell. They want to bring me to the edge of a cemetery, to a fresh grave and say: here lies your son. I don't agree to this," she said.
"I want you, who cared about Alexei, and for whom his death was a personal tragedy, to have an opportunity to say goodbye to him," she added, addressing Navalny supporters in Russia.
Navalny's aide Ivan Zhdanov said Lyudmila's lawyers had filed a legal complaint alleging "violation of the body of a deceased person".
He also offered a reward to members of the security establishment in return for information about what he called Navalny's murder.
The spectacle of Navalny's elderly mother confronting the power and bureaucracy of the state to demand the right to bury her son has moved and angered many inside Russia and beyond.
Tens of thousands have signed petitions to demand that Navalny's body be handed over.
"If you still have any trace of humanity, stop taunting his mother and return the body of her son to her," said Evgenia Kara-Murza, wife of opposition politician Vladimir Kara-Murza, who is serving a 25-year sentence in Siberia on charges including treason, that he has denied.
Journalist Dmitry Muratov, a winner of the Nobel Peace Prize, posted on X, "Just give Lyudmila Ivanovna her son. Just give him back".
Belarusian Nobel literature prize winner Svetlana Alexievich posted: "I want to appeal not only to the Kremlin. I want to ask all people, all of us - keep on saying and saying that they must give back the body to his mother."
Navalny's wife Yulia and her daughter Dasha on Thursday met US President Joe Biden, who called Navalny "a man of incredible courage" and has said he holds Putin responsible for his death.
