The 23-year-old actress – who previously dated DJ Samantha Ronson – has grown close to photographer 36-year-old Indrani since they met on a photo shoot last autumn.
Indrani – also known as Julia I. Pal-Chaudhuri – told the New York Post newspaper: "We have been spending a lot of time together. I have never had a relationship with a woman before, but Lindsay is just somebody who I find fascinating, gorgeous and extremely smart, as well as super-hot.
"Lindsay gets a lot of bad press, but she's a really strong, creative woman and is trying really hard to get her life in a good, positive place."
The couple have been on a series of dates in Los Angeles and recently spent the night together at the actress' hotel.
Indrani's photography partner, Markus Klinko, thinks the pair are good for each other.
He said: "Lindsay and Indrani have been seeing each other since we shot her last fall. I've seen them on dates, I have seen them making out.
"Indrani is a good influence on Lindsay. She is the opposite of a party girl - a Princeton graduate, she's into art and is a philanthropist - not what you'd expect the typical girl for Lindsay to go out with.
"When they are together, they talk about art and the deeper meaning in life."