The former England boss has been in charge since taking over from Dick Advocaat in July last year, and could now remain at the helm for the 2018 FIFA World Cup, which Russia is scheduled to host.
Sports minister Vitaly Mutko claimed a deal was on the verge of being signed, with the 67-year-old's son Pier Filippo Capello expected in Moscow next week to finalise the agreement.
"I can say that an agreement in principle between the coach and the RFU (Russian Football Union) has been reached," Mutko told local media.
"As far as I know, his lawyer will come to Moscow on November 21-22 and everything should be settled.
"There are several options for the date of the signing of the new agreement but probably everything will be officially formalised after the draw for the World Cup finals (on December 6)."
Mutko feels an extension of Capello's contract will benefit the national side, and added: "We cannot lose the momentum and the continuity now."
Russia are unbeaten in five fixtures and face South Korea in a friendly on Tuesday.