The Red Devils are on the lookout for a new shot-stopper to provide competition and cover for David de Gea and had sounded out the German as a target.
Trapp helped the Bundesliga side lift last season's Europa League trophy and, despite initially mulling over a switch to the Premier League, he has opted to stay put.
Speculation regarding Trapp's future was so fierce that the 32-year-old announced his decision via a message on the club's twitter account.
Trapp said: "There have been a lot of reports about interest from Manchester United.
"It's true that a written offer was tabled. Since it came from such a world-famous club, of course I considered it.
"Yesterday I informed the management of both clubs that I've decided to stay at Eintracht. We've made history here together and experienced some unforgettable things. I have absolute faith in us!"
Unlike United, Trapp will turn out for Eintracht in the Champions League following their Europa League success against Rangers in May.
The German made a return to Eintracht in 2019 after initially leaving the club in 2015 to sign for Paris Saint-Germain, where he spent four years.
Trapp was part of the PSG side that were beaten in the infamous 6-1 Champions League Round of 16 clash against Barcelona at the Nou Camp in 2017.
With the door now firmly closed on a deal for Trapp, United have wasted no time in turning their attention elsewhere with Newcastle United's Martin Dubravka a possibility.
The Slovakia international has fallen down the pecking order at St James' Park after the summer arrival of Nick Pope.
But Dubravka was then left out of a second-string Newcastle side that overcame Tranmere Rovers in the Carabao Cup on Wednesday night.