A publicist for the "Bourne Identity" actor told the magazine that after dating for two years, Damon popped the question "shortly before Labor Day."
Damon's publicist, Jennifer Allen (search), didn't immediately return a call from The Associated Press late Monday.
The couple haven't set a wedding date, Us Weekly said. It would be Damon's first marriage. Barroso has a daughter, Alexa, from a previous relationship.
Damon's screen credits also include "The Brothers Grimm," "Ocean's Eleven" and "The Talented Mr. Ripley."
He shared a screenplay Academy Award with his friend Ben Affleck for 1997's "Good Will Hunting."