Murray, who retired in 2024, previously had a coaching partnership with Novak Djokovic that ended in May 2025.
Former US Open semi-finalist Draper has struggled with multiple injuries this year and will miss the French Open due to ongoing knee issues.
"I am very grateful for everything Jamie Delgado has done for me over these past six months. He is a world-class coach and a great man," Draper said in a statement to British media.
"In the interim, I will continue to be supported by the excellent team at the (Lawn Tennis Association), with the addition of Andy Murray, who will be supporting me throughout the grasscourt season," he added.
Draper, 24, has never gone beyond the second round at Wimbledon, and only one of his three ATP titles was won on grass.
He did not specify what role Murray would take on his team, though some media reports said it would be in the coaching setup.