Delgado has been in Murray's team since 2016, helping him to a second Wimbledon title and the world number one spot in the same year.
He took over as the main coach from Ivan Lendl in 2017, though much of his time with the Scot has been spent rehabbing, with Murray's career derailed by hip injuries.
He has now left Murray's team to go and work with Canadian Denis Shapovalov, the world number 14 who beat Murray in the third round at Wimbledon earlier this year.
Murray will continue a trial with German Jan De Witt at an exhibition event in Abu Dhabi next week as he prepares for next month's Australian Open.
De Witt, who counts Gilles Simon, Gael Monfils, Tatsuma Ito, Nikoloz Basilashvili and Viktor Troicki as former players he has worked with, spent a few days working with the former world number one at the National Tennis Centre in Roehampton this week.
Murray, who was trialling with Johanna Konta's former coach Esteban Carril at a tournament in Stockholm last month, will play Dominic Thiem at the Mubadala World Tennis Championship, with a match against Rafael Nadal if he beats Thiem.
Murray is currently ranked 134th in the world as he continues to try and reignite his career after having a serious hip surgery in January 2019.
British number one and US Open champion Emma Raducanu is also in action at the event in Abu Dhabi.