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."