Barnsley boss Danny Wilson is looking to bolster his attacking options after seeing Chris O'Grady leave for Brighton earlier this month and he has targeted a move for Murphy.
Murphy impressed at Dagenham last season after arriving from Dutch side Telstar on a free transfer last summer.
The 23-year-old scored 13 goals for the Daggers last term and his performances caught the eye of several clubs higher up the football pyramid.
Former Arsenal youngster Murphy turned down a possible move to Peterborough in January to stay at Dagenham and the League Two club know they face a fight to keep hold of him this summer.
Murphy is entering the final year of his contract at Dagenham, and Barnsley are hopeful of luring him away to Oakwell as they look to build a squad capable of winning promotion back to the Championship