The Geordie duo told Simon Cowell someone else could have their job unless things on the show changed.
Ant and Dec say they nearly quit Britain's Got Talent over concerns with how the show was being edited.
The 44-year-old presenters say they only decided to stay after a make-or-break meeting with Simon Cowell in LA, where he promised to sort out the problem.
The revelation comes in the Geordie presenting duo's forthcoming autobiography, Once Upon A Tyne, which has been previewed in The Sun.
Anthony McPartlin and Declan Donnelly say they feared they were being side-lined and thought their contribution was not being reflected in the edited shows.
After talking to ITV bosses, a crisis meeting was set up with Cowell at an Italian restaurant in Los Angeles, California.Advertisement
In the extract from the book, Ant says: "We told him everything - that we weren't being used, that we could be doing other stuff, that maybe someone else should take over."
They also said Cowell's shoddy timekeeping, often turning up late to auditions, "drives us nuts".
Ant went on: "To be fair to Simon, he sat there, took it all and listened intently for four, maybe five cigarettes."
Dec said Cowell quickly offered a solution, saying: "He made a promise to us: that things would change and that we'd never feel like that again.
Back from our break and look what’s been delivered... our first copy hot off the press! 📖 Just one more week until you guys can get your hands on it too. Preorder now at https://t.co/4CQDMwGjMM #OnceUponATyne pic.twitter.com/D6nWJuJr4L
— antanddec (@antanddec) August 27, 2020
"He desperately wanted us to stay and we told him we'd do the next series and see how it went - and, to be fair to him, he was as good as his word, things did change."
After the meal, Cowell, 60, is said to have offered them a lift back to their hotel in his chauffeur-driven Rolls-Royce.
Ant and Dec have hosted Britain's Got Talent since its launch in 2007 and have continued on the show since their ultimatum.
@ApplauseStoreUK do we know if anyone has managed to get on to the #BGT virtual audience or are we just the unlucky ones?
— Steve Buckley (@Buckley1984) August 28, 2020
This year's show has been split into two parts due to coronavirus, with the audition episodes running in April and May, and the live shows due to start on 5 September.
Due to social distancing restrictions, the live shows will take place without a live studio audience.
Cowell has full ownership of both Britain's Got Talent and The X Factor, after buying Sony Music Entertainment's stake in both earlier this year.
He also gained ownership of all the international versions of both programmes.
Last year, Cowell signed a new five-year deal with ITV which will see his shows appear exclusively on the channel until at least 2024.
The agreement includes five series of Britain's Got Talent and at least one of The X Factor.
Ant and Dec are also gearing up for their next high-profile ITV presenting gig - I'm A Celebrity… Get Me Out Of Here - which this year will take place in winter, in a derelict castle in Wales.
ITV boss Kevin Lygo confirmed at this year's Edinburgh TV Festival that most of this year's celebrities have already been booked for the rebooted show.
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