Murty will take charge of his fourth match on Wednesday when Rangers host St Johnstone after falling nine points behind second-placed Aberdeen in the Ladbrokes Premiership following consecutive defeats by Dundee and Inverness.
The Ibrox board have been interviewing candidates to replace Mark Warburton on a long-term basis but their first appointment looks set to be for a new director of football role.
They hope to confirm that position later this week, with Southampton’s head of scouting and recruitment Ross Wilson understood to be their preferred choice.
A manager could follow relatively quickly but Murty looks likely to be in charge for Saturday’s William Hill Scottish Cup clash with Hamilton, and the following game is away at Celtic on March 12.
“I think we have to respect the club’s wishes and make sure they get it right,” the club’s under-20s coach said.
“If we appoint quickly, and get it wrong, it would be much more damaging. So we have to be respectful of the board’s wishes, to go through their process, do their due diligence, and get the right person in place.
“I’ve been told the process is ongoing, interviews are taking place, and someone will be appointed. I don’t know when it will be.”
Murty has admitted he is not enjoying being thrust into the hot seat, although he is relishing the chance to gain experience.
“It’s not about me, it’s about the football club and the players,” he added. “I’m here to facilitate the players playing well, hopefully, and making sure they know what they need to do on a match day. I’m here fulfilling the role I was asked to do.
“I’m content to look in the mirror and say that the practices we put on, the work that we’re doing, is of a good standard, it’s what the players need.
“If I felt I wasn’t being impactful to the players, I would feel more than comfortable going to the board and saying: ‘You need to replace me’.
“As it happens, the board have been fantastically supportive, all they are asking me again is: ‘Can we help you?’
“But I think if you look at the Inverness game from the Dundee game, there was a definite impact, the players responded really, really well and there was an improvement in our performance levels.
“I reiterate however that we need to get points on the board to make sure that all people see that this is a positive and a step forward for the football club.”