Hughes, who joined Liverpool from Bournemouth, informed the club's ownership of his desire to seek a new challenge after delivering the 2024/25 league title and Champions League qualification the following season.
"Richard joined us at the start of a period of change, embracing that responsibility by leading our football operations department and overseeing recruitment while also providing valuable support to our coaching staff," Fenway Sports Group president Mike Gordon said.
"The results we achieved in these areas during this period are testament to this contribution. Nevertheless, we respect Richard's life choices and wish him and his family well in their next venture."
Hughes appointed Arne Slot as Klopp's successor, a decision that paid immediate dividends with a Premier League title triumph in the Dutchman's debut campaign.
He also presided over an ambitious recruitment drive, bringing in high-profile signings such as Florian Wirtz, Alexander Isak and Hugo Ekitike among others as Liverpool invested heavily to compete at the highest level.
His tenure was not without controversy, however, with the squad criticised for lacking balance and depth in certain positions despite the significant outlay.
Liverpool endured a more difficult 2025/26 season, leading to questions over some of Hughes' expensive recruitment decisions and Slot was sacked when their title defence turned into a scramble for Champions League qualification.
Hughes, who has been linked with a move to Saudi Pro League side Al Hilal, expressed gratitude for his time at Anfield.
"I would like to reiterate what I said upon my arrival – it has been a source of great pride to have been given the opportunity to serve this incredible and unique institution. That is something I will always be grateful for," he said.