The hopes of Russian athletes competing at the Rio Olympics now rest with Olympic chiefs giving them a special dispensation.
Athletics' world governing body the IAAF has maintained its doping ban on all Russian athletes.
Russia was suspended from all track and field by the IAAF in November after an independent report from the World Anti-Doping Agency revealed widespread state-sponsored doping.
The vote by the IAAF should in theory be decisive, but the IOC, concerned about innocent athletes being punished, has not ruled out granting some Russians a special dispensation when it meets next week.
A task force led by Norwegian Rune Andersen was set up to monitor progress in reforming Russia's anti-doping programme, and in March reported that there was "significant work to do".
Earlier this week WADA released another damning report on the doping situation in Russia, one of the world's sporting super-powers, who were second behind the United States in the athletics medal table at the 2012 Olympics.
That report revealed 52 new failed tests and stories of extraordinary attempts to avoid, obstruct or intimidate drug testers, suggesting that attempts to change the culture of doping in Russia had failed.
The IAAF Council had been due to hear from Andersen again on Friday, and from Russian representatives.
The Federation had scheduled the vote for June so that, if the ban were to be lifted, Russian athletes would have a reasonable chance to register Olympic qualifying standards before the July 11 cut-off.
International Olympic Committee (IOC) President Thomas Bach was asked this month if the IOC, holding its own Olympic Summit on June 21, would be prepared to overrule an IAAF ban and allow Russian athletes to go to Rio.
He replied: "I cannot speculate."
"This meeting on the 21st will be to protect the clean athletes and ensure a level playing field for all the athletes participating in Rio," he said.
Russian President Vladimir Putin said there should be no collective punishment for Russian athletes and that doping should not be politicised or used to push an anti-Russian agenda.
"Of course, it is unfair," Putin told editors from leading international news agencies at a supper on the sidelines of the St. Petersburg International Economic Forum.
"Responsibility must be always personalized. If someone in your family has committed an offence is it fair to prosecute all the family members ... the people who have no relation to the violations - why should they suffer for others?"
He added that this does not fit into "any framework of a civilized behaviour" and that he hoped a solution would be found.
Russia also revealed an open letter sent to the IAAF saying that Russia had met all the conditions asked of it, including overhauling its disgraced athletics association and introducing additional testing.
"Clean athletes who have dedicated years of their lives to training and who never sought to gain unfair advantage through doping should not be punished for the past actions of other individuals," Mutko wrote.
Russian athletics official Mikhail Butov said the will go over the decision to see if there is grounds for appeal to the Court of Arbitration for Sport.
He was scornful of the IAAF's decision that athletes who trained outside Russia and could show they were not tainted by the Russian system might still be able to compete at the Rio Olympics as neutral competitors asking how they were supposed to prove that.