Wright, 45, was spotted with the 31-year-old on a close dinner date in Australia earlier this week while she was filming The Grandmothers, according to People.
The photos see Wright stroke Foster's face, while another shows Foster kissing her hand.
Robin Wright was previously married to Sean Penn for 14 years, before their divorce was finalised in 2010.
Ben Foster has been linked to playing Bruce Willis's son in Die Hard 5.
He will also appear in Prometheus and Gotti.
Watch the trailer for Rampart below: