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."