The 41-year-old actress is good friends with the 34-year-old 'Homeland' star, and since they both have similar facial features and are petite blondes, she is hoping they'll have the chance to team up on screen and play sisters.
A source told RadarOnline.com: "Gwyneth has known Claire personally, but casually, for almost 20 years and when Gwyn was in her early 20s and Claire was a late teen actress, they both became famous around the same time in the mid-90s.
"Much has been made of the pair's similar looks and Gwyneth has finally decided that she and Claire have to find a project where they can play sisters!"
Gwyneth is said to be keener on the idea while Claire, who is currently starring as CIA agent Carrie Mathison on the third season of hit TV series 'Homeland', is taking things "one day at a time".
The source added: "But for now, this seems like a bit of a one-way street.
"Claire is a cool customer and doesn't get all worked up about 'dream projects' the way Gwyneth does.
She's a 'one day at a time' kind of girl, so no one knows if Gwyneth's dream will come true."