Clement was relieved of his duties early on Thursday, barely nine months into the role.
The Royals currently lie 21st in the English Championship, and are only above the relegation zone on goal difference.
Former Reading right-back Murty has been linked with a return to the Madejski Stadium, where he is still adored by fans for his contribution to the record-breaking Championship title win in 2005/06.
Murty was touted for the job after Jaap Stam was sacked, and prior to Clement’s appointment.
The former caretaker Gers boss is still at Ibrox, where he oversees the club’s youth programme, but could he be tempted by a return to Reading?
Elsewhere, former Rangers midfielder Charlie Adam has called on Alfredo Morelos to “sort himself out” - or face being pushed out of Ibrox.
Speaking to talkSPORT Adam said: “His ill-discipline is costing him and costing the team. There’s got to be a point where the club and Steven have to go and say ‘sort yourself out or you won’t play’.
“He’s their best player, he’s scored goals but he’s got to learn quick. Eventually Steven will say ‘we have to get rid of him’.”
Morelos was sent off for the third time this season - albeit one red card was rescinded - in the 1-0 home defeat to Aberdeen.
Meanwhile, Rangers’ new club shop in St Enoch Square is close to opening, following reports from eagle-eyed passers-by that signage had gone up on the former Maplin electronics store.
Simply called “Rangers Glasgow” the storefront is white with blue lettering and is expected to open in the not too distant future.