"Lindsay's long been obsessed with the royals, particularly Prince Harry, who she thinks is a perfect match for her," a source said.
"But Eugenie's quite low-key. The two run into each other at their favourite restaurants - La Cenita and bubby's - where Lindsay makes a beeline for Eugenie and invites her to her house," it added.
It is claimed Lohan hopes a royal friendship would help boost her tarnished image.
"She's practically stalking the princess in a desperate bid to upgrade her image," the source said.