The fight – scheduled for 14 April at London’s O2 Arena – is now set to take place on 23 June at the same venue.
“I’m gutted for Martin and I’m gutted for the fans,” said Saunders. “I felt my hand go during a session this week and sought medical advice immediately.
“We’ve been left with no choice.”
Saunders, 28, had been hoping to build on his third successful title defence against Canadian David Lemieux in December.
He will now make the fourth defence of his belt against 35-year-old Murray, from St Helens, on the June bill which will also feature Anthony Yarde, Daniel Dubois, and Ohara Davies.
“This is an extremely tough sport and these things can happen,” said Saunders. “It is vital that I’m firing on all cylinders heading into this fight but for now it’s all about recovering properly and focusing on the new 23 June date.”
Promoter Frank Warren said: “It is extremely frustrating as tickets were going very well and we were all geared up for a fantastic evening of boxing on 14 April.
“I apologise to all of the fans that have bought tickets for the 14 April date but all original tickets will be valid for the new 23 June date.”
Warren said he is also “working on a new date” for the WBO super-lightweight title fight between England’s Terry Flanagan and American Maurice Hooker.
Manchester boxer Flanagan, 28, was aiming to win a world title at a second weight after previously holding the lightweight title from 2015 to 2017.
“I am speaking with Hooker’s representative Dino Duva and we will be making an announcement this week on where and when this fight takes place,” he said.
“It is not straightforward as this is an ordered fight for a vacant title but we are working hard to get everything sorted and our priority is to secure Terry his chance to become a two-weight world champion”.