The Red Devils were linked with several high-profile midfielders over the summer, but ended up completing just one piece of business on transfer deadline day for Everton's Marouane Fellaini.
De Rossi was among those reportedly targeted by United boss David Moyes once his options started to dwindle, with the Italy international having been linked with a move to the Premier League for some time.
Roma were aware of the speculation surrounding their combative World Cup winner and were prepared to talk if a suitable offer was made, but saw a cut-off point set by the club pass before United made their approach.
Garcia told L'Equipe: "In the summer we talked about his future. He is from Rome, but there were some big clubs looking at him.
"I told him that I could not lose him last minute, so we set a date after which no-one would leave. That is exactly what happened.
"A few days after the first Serie A match of the season, Manchester United called the club and called him, but the date had passed and he didn't move."
Garcia and De Rossi have helped Roma to make a perfect start to the 2013/14 campaign, but the Serie A outfit are keen to play down title talk just six games into what will be a long, hard season.
Garcia added: "Juve and Napoli remain the favourites.
"Here, people get worked up very quickly and start talking about titles after just four matches. That is not sensible.
"We won't dupe people. The objective is to get back into Europe."
@varicoseregent: Do you really think Moyes had any direct involvement in this summers transfer negotiations? If he did, then he shouldn't have. And f he did, then Woodwood is even more responsible for the repeated ball ups. I'd have thought Moyes would have had enough on his plate when he joined us without also having to sit around a table hammering out the finer points of transfer deals. And why should a football manager be expected to do this anyway? They're no more qualified to conduct multi-million pound business transactions than negotiators (Woodwood) are to manage a top flight football team! Footballs moved on. Transfers are, by and large, no more about chummy relationships and done by canny managers. They're serious, mega money, international legal ranglings. It would be negligent to let [most] managers deal with these in detail. Especially when they're supposed to be managing a football team.- trumpton