United ended up 35 points behind champions City, who pipped Liverpool to the title by a point, as the Red Devils registered their lowest-ever Premier League points tally.
City boss Pep Guardiola has now claimed four titles in five years, while Liverpool manager Jurgen Klopp can complete a treble of cup trophies this season should the Reds see off Real Madrid in Saturday's Champions League final.
Guardiola and Klopp -- the managers of United's two biggest rivals -- have consistently battled it out for the major trophies, leaving the Red Devils, without silverware since 2017, as bystanders.
However, the Dutchman challenged the suggestion it would take the departure of one or both of Guardiola and Klopp for United to be able to compete again.
"In this moment I admire them both," Ten Hag said as he was unveiled at Old Trafford on Monday. "They play in this moment really fantastic football, both Liverpool and Manchester City.
"But you always see that an era can come to an end. I am looking forward to battle with them."
The former Ajax boss, 52, is United's fifth permanent manager since the club last won a league title in Alex Ferguson's farewell season in 2013.
Ten Hag also had a spell as Bayern Munich's reserve team manager earlier in his career and said he was used to the demands of working at a big club.
"This is a project, it takes time," he said. "I know in this club -- I have the experience of Ajax and Bayern -- in the top (clubs) there is never time.
"We want to win every game so we go from game to game and then we will see. This club has a great history, now let's make the future. It's really exciting to do that with the people around in the club."
Ten Hag refused to be drawn on how many new signings he is targeting in the transfer window, but said there would be a role for top-scorer Cristiano Ronaldo, who turns 38 next season.
Ralf Rangnick's interim time in charge since November failed to summon any sort of response from an expensively assembled squad as United finished 13 points outside the top four.
Rangnick is set to take over as Austria manager but is still contracted to the club in a consultancy role for the next two years.
Ten Hag, though, was non-committal on how much contact he will have with Rangnick.
"That is on the club," he said of Rangnick's role. "I analyse by myself, I observe and I speak with a lot of people, but finally I draw my own line."
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