The Cherries have been long-term admirers of the Crystal Palace forward and had launched an official bid to sign the forward earlier this month.
Eddie Howe’s side were originally hoping to take the former Brighton, Rochdale and Carlisle forward on loan.
But it was understood that Palace would only part with Murray,31, if it was a permanent transfer – and now it appears a deal has been reached, with the move expected to go through before the 6pm deadline on Tuesday.
Murray began his career at Carlisle before being shipped out on loan to Stockport County and then Rochdale, where he signed permanently as a 23-year-old in 2007.
He then moved to Brighton for £300,000 in in January 2008 and went on to score 53 goals in over 100 league appearances for the Seagulls.
Murray enjoyed a somewhat successful loan spell back in the Championship with Reading last season, where he netted eight times.