The 30-year-old was reportedly the subject of an opening offer of £12million from the Premier League champions on Friday morning, with manager David Moyes looking to bolster his midfield.
News of the rejection hands another blow to United's quest to strengthen with Moyes having bids to sign Barcelona's Cesc Fabregas snubbed earlier in the summer.
De Rossi joined Roma's youth system as a 17-year-old in 2000 and has become a pivotal part of Mauro Baldissoni's set-up, with 308 first-team appearances.
"As for De Rossi, we will always reply 'no thank you' to any formal offer that is proposed for him, as would the player," the manager told Sky Sport Italia ahead of Sunday evening's match against Hellas Verona.
"The club will not accept any offers for him. Both Roma and Daniele are coming off a year that didn't go well and we want to make up for that together.
"The club doesn't intend to accept offers for De Rossi and he has no intention of leaving us. Daniele is an integral part of our history."
It is understood Moyes has no intention of making a second offer for De Rossi as his attention turns to securing the signature of Everton's Marouane Fellaini ahead of the 11pm deadline on Monday.