The American arrived at Oakwell in the summer of 2019 after latterly working as technical director at DC United. Prior to that, the former Major League Soccer footballer was the technical director at Real Salt Lake and head scout at New York Cosmos.
Forest are reportedly hoping to tempt Murphy to the City Ground, with current chief executive Ioannis Vrentzos having informed Forest he intends to leave the club.
The East Midlands outfit aim to have his successor in place early next month, with owner Evangelos Marinakis having identified Murphy, 35, as his preferred candidate.
Speaking about his time at Oakwell in a recent interview with The Yorkshire Post following the end of a second eventful - and successful season at the club - Murphy said: "I want to be able to sit down and have time for reflection, but it has been a busy period.
"I cannot be more honest than say that this has been a life-changing experience for the best.
"The fact I have been given the opportunity to come here is an amazing one and is something I count myself as fortunate to have every single day.
“Everyone has ambitions, but certain places are special. If you see continued growth, of course, you want to stay and stick by that.
"Not only in the dressing room and staff, but in the front offices and backrooms away from the bright lights, this is a great place to have a lot of strong relationships that I count on."