Emilia Clarke gets close to Charlie McDowell in Los Angeles

Nov. 6 (UPI) -- Emilia Clarke cozied up to rumored boyfriend Charlie McDowell in Los Angeles.

Popsugar published photos Monday of the 32-year-old British actress kissing McDowell, 35, during a casual outing Sunday in Venice, Calif.

The Game of Thrones star sported an oversized sweater, jeans and sunglasses as she and McDowell ran errands. E! News said the pair picked up groceries at Erewhon natural foods market.

"They walked along the shops and looked to be very cozy and showing lots of affection," a source said.

Clarke and McDowell, the son of actors Malcolm McDowell and Mary Steenburgen, were first linked in October after appearing to celebrate Clarke's birthday together. The pair both shared the same photo on Instagram.

"Well that was a birthday I won't be forgetting in a while," Clarke captioned the picture, which shows the silhouette of two people kissing.

"Happy birthday, E.," McDowell wrote on his own account, adding a heart emoji.

Clarke plays Daenerys Targaryen on Game of Thrones, which will return for an eighth and final season on HBO in 2019. Co-executive producer Bryan Cogman recently told Entertainment Weekly the season will be "haunting" and "bittersweet."

McDowell is best known for directing the film The One I Love starring Mark Duplass and Elisabeth Moss. He previously dated actress Rooney Mara.