De Gea's contract expired on June 30th but at the time the club said they were still in discussions with the shot-stopper over an extension.
The Spaniard, who joined United in 2011 when Alex Ferguson was manager, was the current squad's longest-serving player having made nearly 550 appearances.
"I just wanted to send this farewell message to all Manchester United supporters," De Gea wrote on Twitter.
"I would like to express my unwavering gratitude and appreciation for the love from the last 12 years. We've achieved a lot since my dear Sir Alex Ferguson brought me to this club."
De Gea won the Premier League in his second season with the club - Ferguson's last as manager - while he also won the League Cup, FA Cup and Europa League.
He also won the Golden Glove for most clean sheets in a season twice, the second coming in the 2022-23 campaign as United finished third and qualified for the Champions League.
But the four-time club player of the year had come under the spotlight for shortcomings and errors in recent years.
The former Atletico Madrid keeper was one of the highest-paid players at United according to British media, but he has not featured for Spain since 2020.
United have been linked with a move for Inter Milan goalkeeper Andre Onana (27).
"It's been an unforgettable and successful period since I came here. I didn't think from leaving Madrid as a young boy we would achieve what we did together," De Gea said.
"Now, it's the right time to undertake a new challenge, to push myself again in new surroundings."
