The 33-year-old midfielder, who was the Tigers' club captain last season but was not offered a new deal, becomes the Australian side's marquee player for the 2014-15 campaign.
Slovenia international Koren joins up with Damien Duff, who signed for Melbourne in June after his release by Fulham.
Melbourne head coach John van 't Schip told the club's website: "Robert is someone who we feel can add something extra to the team and can help us get to the next level.
"He is a team player who we believe will contribute well and help our entire squad reach their full potential and help deliver results."
Koren, who has been given Melbourne's number 10 shirt, will join up with his new team-mates for training on Monday.
Source : PA
Source: PA