The Cherries forward, who joined in a deadline day move from Crystal Palace last summer, will make the move along the south coast to the AMEX Stadium to join up with Chris Hughton's squad.
Murray made 23 appearances and scored four goals in his first season under Eddie Howe's tutelage, including a memorable late goal in the 1-0 win over Chelsea at Stamford Bridge in December.
The 32-year-old now joins up with the club he spent three-and-a-half years at from 2008 to 2011.
Murray would not be eligible to face his parent club if Brighton were drawn against AFC Bournemouth in any cup competition during the 2016/17 season.
