But Murty's claims were severely dented on Sunday when Rangers lost 5-0 to Celtic at Parkhead, a result that secured the Scottish Premiership title for the Bhoys.
Over the past week, it has been reported that Steven Gerrard is likely to take over as Rangers manager for next season.
"I was 48, 49 when I joined [Celtic]".
Assistant coaches Jimmy Nicholl and Jonatan Johansson will take over dugout responsibilities for Saturday's home game with Kilmarnock while Rangers chiefs look to lift the dark gloom now sitting over Ibrox by securing the appointment of Kop icon Gerrard.
The past six months has been Murty's second temporary spell in charge.
The club said in a statement: "Rangers are grateful to Graeme, who did not hesitate when asked last October to become interim manager after the departure of Pedro Caixinha and then, at the turn of the year, when he was offered the role as manager until the end of the season".
He returned to Liverpool's academy last summer to take charge of the under-18 side but it seems likely that the 37-year-old is ready to step up to management at a big club.
"The guy went in there and he's done his very, very best in what looks like, from the outside, to be very, very trying and hard circumstances".
Murty's tenure at Rangers has been under severe scrutiny in recent weeks, with Kenny Miller and Lee Wallace having been suspended following an alleged disagreement with the manager in the aftermath of the cup defeat to Celtic.
The Glasgow giants lie third in the Premiership table with three games of the season remaining, with only goal difference separating them from fourth-placed Hibernian.
After the match, Murty did not speak to the media.