The New York Daily News said Lohan and Ronson have been inseparable since they were seen kissing on P. Diddy's yacht in Cannes this spring.
The actress has not publicly commented on the nature of her relationship with Ronson, a 30-year-old British disc jockey and recording artist. However, the media has been describing Ronson as Lohan's girlfriend for weeks.
The Daily News said Ronson bought Lohan the expensive bauble as a birthday gift while they were out in Los Angeles window shopping.
"Sam took Lindsay shopping to celebrate her 22nd birthday," a source told The Daily Mirror. "Although Lindsay is the major breadwinner, Sam has her own money and is very proud. ... It's not an engagement ring -- simply a token of her commitment."