A source is said to have confirmed to PEOPLE magazine that George, 52, popped the question "not that long ago", although the pair haven't started planning their big day.
"George and Amal are trying to keep things very low-key but they also aren't really trying to hide this, it doesn't seem," said the source. "I think it's like they want the people they love to know that this is real, that they plan on being together forever."
The news of George and Amal's engagement comes several days after the couple were spotted dining at sushi restaurant Nobu, where Amal was seen flashing a "huge ring" on her ring finger.
Say HELLO! to George Clooney's new girlfriend Amal Alamuddin
George and 36-year-old Amal were joined at the Malibu eatery by Cindy Crawford and her husband Rande Gerber, who are long-time friends of the Kentucky-born actor.
This was not the first famous double date for George and Amal. In March the couple were seen at LA's Kazu Sushi with Emily Blunt and her husband John Krasinski.
The foursome appear to be close friends as they were seen hugging each other outside the restaurant before parting to go home.
Beirut-born Amal was first linked to George in October and have enjoyed a series of dates around the world including a safari in Tanzania, a holiday in the Seychelles and a screening at the White House.
A graduate of both Oxford University and New York University of Law, Amal specialises in human rights, international law and extradition for the UK's prestigious Doughty Street chambers.
One high-profile client is WikiLeaks founder Julian Assange, whom she's represented in extradition proceedings with Sweden.
Fluent in English, Arabic and French, Amal has also been appointed to various UN commissions and is an advisor to former UN Secretary-General Kofi Annan on Syria.