'Very important' for Man Utd to get right sporting director: Ten Hag
United announced this week that Murtough was stepping down from the football director role after 11 years at Old Trafford in a variety of positions.
With INEOS in control of United's football operations since Jim Ratcliffe became a co-owner in February, the British billionaire is looking at moves for Newcastle sporting director Dan Ashworth and Southampton director of football Jason Wilcox.
United boss Ten Hag acknowledged the significance of finding the ideal candidate for the post.
"You mention the new season, it's very important, the new ownership is working on this and we have to replace this function to go into the new season," Ten Hag told reporters on Friday.
"For the moment, of course I miss his support, but he moved on, there have been choices made. We work very good together, so I say thank you to John and I wish him all the best for the future.
"Also I work with the new ownership very good and closely together and that will not change. It doesn't have an impact on the way I can work here.
"Conditions were perfect and are still perfect, so I'm happy with that, but we want to set the right conditions to be successful."
United travel to Bournemouth this weekend looking to avenge their 3-0 home defeat in December.
Ten Hag's sixth placed side have little chance of qualifying for next season's Champions League via a top-four finish.
And a repeat of that dismal loss to the Cherries would pile further pressure on the beleaguered Dutch coach.
"Tomorrow will not be different, that is the way they play, they want to fight with you. So it's about making sure you go in that fight, have the belief and you need to support each other to win the battles, to outplay them and to outrun them," Ten Hag said.
United are assessing the fitness of Marcus Rashford after the England forward came off with an issue in the 2-2 draw with Liverpool last weekend.
Scotland midfielder Scott McTominay, an absentee for that game, is also being checked by United's medical staff ahead of Saturday's clash at the Vitality Stadium.
United also have a rash of defensive injuries to Raphael Varane, Jonny Evans, Lisandro Martinez, Victor Lindelof, Luke Shaw and Tyrell Malacia.
"It's very difficult. Everyone who has a little bit of knowledge about football, they know it will affect your results and performance," Ten Hag said.
"Back four is always a foundation to getting the results and we didn't have that.
"But then still, we have to win and we know that. Others have to take the responsibility. But when you oversee it, you know also it has a negative impact."