The Cherries have been chasing Murray for some time and originally hoped to take the former Brighton, Rochdale and Carlisle forward on loan.
But reports earlier this week suggested that Palace would only part with Murray, 31, if it was in permanent transfer, with claims on Friday suggesting Bournemouth had now firmed up their interest with an offer in the region of £5million.
Eagles boss Alan Pardew said on Thursday that Murray was “our player” and that he owed the club for giving him his chance in the top flight.
Murray began his career at Carlisle where he was 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 apperances for the Seagulls.
Murray was an bit-part player for Palace last season after returning from a somewhat successful loan spell back in the Championship with Reading, where he netted eight times.