Gers have been without a permanent manager since Pedro Caixinha was sacked in late October, with Murty taking temporary charge at Ibrox.
Murty initially started promisingly, with back-to-back wins over Hearts and Partick Thistle, but successive defeats have since highlighted the need for clarity over the future at Ibrox.
Derek McInnes has committed his future to Aberdeen following links to the role, while another favourite, Alex McLeish, insists he has not been approached despite one bookmaker on Monday suspending betting on his appointment.
Murty admits there has finally been movement in the search, but has stressed the importance of Rangers taking their time to get the decision right.
He said: “I’ve discussed with Mark Allen and I’ve discussed with Stuart Robertson (on the Gers board) that this process is ongoing.
“There has been movement but we’re not quite there yet, and I think that to get the next five, six, seven years right, we need to take our time now to get this appointment right.
“If we take a little bit more time now to get this appointment right, we actually get the next five years of progress and secure the future of this football club at the top of the league, and that’s crucial.”