The 37-year-old is currently undergoing a medical ahead of a move to the Russian Premier League side, where it is reported he will sign a two-year deal.
It is understood that Spartak moved for Terry after French centre-back Samuel Gigot was ruled out for the season with a serious injury, with the Chelsea legend available on a free transfer after his contract at Villa expired.
A return to Villa was also touted for the former England defender, having recently said that he feels there is "unfinished business" in Birmingham.
"I loved my time at Aston Villa," he told the Daily Mail . "I am still devastated that we didn't go up.
"The manager, players and supporters were incredible with me but I hate losing and I feel there is unfinished business there.
"Steve Bruce is a great man and with the backing of the new owners, Steve will hopefully get the club back into the Premier League.
"Don't get me wrong, if I don't play again I am content with what has been an unbelievable career, but I'm still hungry to go and play if everything around it is right. If it's right for me and my family."
The Russian Premier League side finished third in the table last season although failed to qualify for the group stages of the Champions League after losing out to Panathinaikos in the third qualifying round.