De Gea's contract with the 20-time English champions expired at the end of June, and it was confirmed earlier this month that his time with the club had come to an end.
The 32-year-old made 545 appearances for the Red Devils in all competitions, including 58 last season, and he won the Golden Glove for the 2022-23 Premier League campaign, keeping 17 clean sheets.
Andre Onana has now arrived to become the club's number one goalkeeper, with the Cameroonian completing his first training session on Friday after arriving from Inter Milan.
"I think David had a huge contribution on our level of performance last year and our result. I want to mention all the clean sheets he made but it was also always a team performance because we defend with 11," De Gea told reporters during his press conference on Friday.
"Not only with the goalkeeper but it was absolutely obvious that David played a huge role and a huge contribution in that. But there is coming a time for every player, that a club is going to make changes. It was the right moment in our perspective."
Ten Hag also insisted that De Gea will have the chance to "say farewell" to the supporters at some stage.
"Yeah but he will get that. Big contribution, he deserves it, if it is up to me. David after 545 games, all his contribution, after all his performance over this period," he added.
"It is massive what he did. For the rest of his life he is a Manchester United [legend]. We have to say farewell in front of all of our fans at Old Trafford, absolutely."
Ten Hag was also asked about Onana's playing style in contrast to De Gea, with the Cameroonian well-known for his ball-playing abilities, which is an area that the Spaniard struggled in last term.
"I expect him to do exactly that. I think Andre is a goalkeeper who can play behind a high block but also a very low block. Therefore it is so good he is now in," added the Dutchman.
"But the co-operation is so essential so we need players in pre-season then we can work on that. But I am convinced he will adapt and the players will also be aware and take advantage of his qualities.
"You have already seen with different players like Matej [Kovar] he did very a well. Our players will very quickly adapt to that and take benefit from it."
Man United have beaten Leeds United and Lyon in their two friendlies this summer, and they will officially begin their pre-season tour of the United States against Arsenal on Saturday night.