"Harry (Morton) has been to Jen's Beverly Hills house and they have had several restaurant dates as well," the source told People. "Just like Jen, Harry is a dog lover and they both love the beach."
Morton, 29, is the Pink Taco restaurant chain founder and Hard Rock heir who dated Lohan for about three months in 2006.
The magazine said Aniston, 41, also was seen recently with "Cougar Town" actor Josh Hopkins, but friends are rooting for Morton, the source says.
The source tells People, "Jen likes Harry because he is kind and low-key. Even though Harry is very successful and wealthy, he is not flashy and just like Jen, has a circle of close friends that are protective of him."