TMZ reports that Drew was recently spotted leaving a medical office in the company of her fiancé, holding a sonogram.
The pictures were taken last week in Beverly Hills.
Many assumed that the 36-year-old is pregnant with her first child.
If the rumors prove true, the actress will have to put her career on hold. However, her rep has yet to comment on the allegations.
Barrymore and Kopelman, a 34-year-old art consultant, got engaged at the end of last year.
This will be the third marriage for Drew. She was previously married to bar owner Jeremy Thomas in March 1994 and to comic Tom Green in July 2001.