Defence lawyer Muhsin al-Suwaidi said Anatoly Bilashkov and Vassily Pokchov will be told of the court's decision on 30 June.
Al-Suwaidi confirmed that the prosecutor general had called for the pair to be executed during Tuesday's hearing.
Both are charged with murdering Salim Khan Yandarbiyev, attempting to murder his 14-year-old son, Dawud, and smuggling arms into Qatar.
Anatoly has pleaded not guilty to the 13 February assassination of the retired Chechen official living in exile, but Pokchov has admitted responsibility.
The closed-door trial began in Doha on 11 April and 13 prosecution witnesses have been heard – including of a passerby who noted the registration number of a suspicious Russian vehicle on the day of the attack.
In Moscow, the Russian Foreign Ministry said it was continuing to work on behalf of the pair.
"We are doing everything possible so that our citizens return home," Mikhail Troyansky, an official with the ministry's press service, told news agencies and Moscow Echo radio.
The murder sparked a diplomatic row between the Gulf state and Russia.
Qatar expelled the first secretary of the Russian embassy in March after detaining him along with the agents in connection with the car bombing.
The assassinated exile, who briefly headed Russia's war-torn separatist republic of Chechnya in the mid-1990s, had lived in Qatar for nearly three years with his family.