Rivers sustained the injury during the third term against the Lions while he was scrambling for the ball in a pack.
"Jared unfortunately jammed the surfaces of his right elbow together and tore the medial ligament. He'll be fully training in three to four weeks and be ready to play shortly after," Melbourne club doctor Andrew Daff told melbournefc.com.au.
But Rivers is remaining optimistic that he can return for Melbourne's round one clash against Carlton at Telstra Dome.
"My aim is to get back by round one, but we'll see how it goes," Rivers said at Trevor Barker Oval in sandringham where the Demons were training on Monday morning.
"They said between four-to-six (weeks), but it just depends on how I treat it and I'll be doing plenty of physio to try and get back for round one.
"I'm glad I did it in the pre-season, rather than the (premiership) season.
"I've never done an elbow before and I thought I might have hyper-extended it, but I thought it was pretty bad, considering the pain I was in and it's still pretty painful at the moment, but I'm getting plenty of physio and we'll see how we go."
The injury is an unwelcome setback for Rivers, the 2004 NAB AFL Rising Star Medallist, who had set himself for a big season after an indifferent 2005 which was marred by a two-match suspension and a late-season hand injury.
"I had a really good pre-season this year and I thought it was my best pre-season since I've been at the club, so I've got to try and get that past me and hopefully look forward to the season," Rivers said.
Rivers said after a "couple of days off" he would be back running again.
"I've got to wear a brace for three weeks, so I've got to try and not suspend it at all," Rivers said.
In a further blow for the Demons, rookie Matthew Warnock will also spend the next six to eight weeks on the sidelines after breaking his thumb against the Lions.
Warnock will have surgery on Wednesday.
