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, 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."