The American left former club Barnsley at the end of last month after two seasons at Oakwell.
The club named Khaled El-Ahmad as his replacement late last week.
El-Ahmad joins from the City Football Group (CFG), where he has been in post since 2016 and will become the first Swedish CEO within British football following a period of transition.
On moving to Forest, Murphy - who has replaced Ioannis Vrentzos as CEO, said: "I am deeply honoured to be appointed CEO of Nottingham Forest
“It is a historic club with global recognition. It is humbling that Mr (Vangelis) Marinakis has placed his trust in me to lead the club forward. It is a unique opportunity and I can’t wait to get started.
“To the supporters, I look forward to meeting you and representing your club. It has been such a difficult time for you and, as I can already see by your unwavering support in renewing season cards, this is a very special club.
“We are on the precipice of something really tremendous and this city deserves it with its’ loyal fanbase and fantastic supporters globally.
“However, the work starts now. We have transfer business to conduct with Chris Hughton who is an extremely well respected, experienced and successful manager at the highest levels of the game."
Forest chairman, Nicholas Randall, QC, said: “We are delighted to confirm that Dane has joined us today in the role of CEO. He was the outstanding candidate with an impressive track record in the game.
"His work at Barnsley was widely recognised within football as being exemplary and his role in turning around the fortunes of the club in such a short space of time is testament to his vision and talent.
“His broad experience within the game as both player and executive provides him with an insight which will be of huge benefit to Nottingham Forest.
"He is also a man of great character and I am personally delighted at news of Dane’s arrival and everyone at the club is looking forward to working with him."