The dangerous winger has missed the last two games of the Canadiens five-game losing streak with what was called by the team an "upper body" injury.
"They said two or three weeks, so we'll start from there," Kovalev said Thursday.
He said the elbow has bothered him for a couple of months and he was taking painkillers before games, but decided to rest it in hope of coming back stronger for the end of the NHL regular season.
"It wasn't too bad in the beginning and then it got worse and worse," he said.
Critics who suggest he is simply miffed at being benched for most of the third period of his last game, a 5-3 loss to Ottawa on Saturday night, are off base, he insisted.
"It's not that," he said. "I figured I'm not helping the team much with the injury I have and it's better to let somebody else healthy play for me.
"It will be a good thing to rest. If people want to believe something, they'll believe it."
Kovalev, one of Montreal's big guns on attack, has only 13 goals and 25 assists in 57 games this season.