Clooney, 52, and Alamuddin were first spotted together publicly in October last year, reported the Huffington Post.
"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. I think it's like they want the people they love to know that this is real, that they plan on being together forever," a source said.
The pair have apparently been getting serious in recent months and were most recently seen at a special screening of Clooney's movie 'Monuments Men' at the White House in February.
The Hollywood hunk was previously married to Talia Balsam and has since had notable relationships with Sarah Larson, Elisabetta Canalis and Keibler.