Earlier in the campaign, Murty was placed in interim charge of the Glasgow outfit until the end of the season but despite winning 18 of his 29 matches at Ibrox, it appears unlikely that the 43-year-old will remain in the dugout.
Reports have suggested that there has been discontent in the dressing room and that Liverpool legend Steven Gerrard has been approached over replacing Murty, and Rodgers feels that the Englishman has been "thrown to the garbage".
Rodgers told talkSPORT: "Graeme Murty has been put in a position – he was Under-20s coach last year and now he's the first-team manager – and everyone was talking about him being a great guy and was doing very well.
"Now, after a couple of bad results he's been thrown to the garbage and I think it's very discourteous. The guy went in there and he's done his very best in what looks like, from the outside, to be very trying and difficult circumstances."
On Sunday, Celtic recorded a 5-0 win over their Old Firm rivals to seal a seventh successive league title.