The Northern Irish presenter has hosted the Sunday edition of the long-running BBC show since taking over from Adrian Chiles in 2010.
Mark Chapman - who has also occasionally presented the programme - will take over as lead host next season, according to The Sun.
Murray is said to have received complaints from colleagues over his criticism of players on the programme.
A BBC spokesman said: "Colin will continue to play a key role in BBC output."
Match of the Day 2 moved from BBC Two to BBC One last year for the 2012/13 season.
Adrian Chiles hosted Match of the Day 2 when it launched in 2004, with the show aiming to offer a more "light-hearted look" at the Premier League.