At her pre-Wimbledon news conference Saturday, Sharapova was asked about a recent Rolling Stone article where the author surmised that critical comments directed at an unnamed player by Williams were referring to Sharapova.
"At the end of the day, we have a tremendous amount of respect for what we do on the court. I just think she should be talking about her accomplishments, her achievements, rather than everything else that's just getting attention and controversy," Sharapova said.
"If she wants to talk about something personal, maybe she should talk about her relationship and her boyfriend that was married and is getting a divorce and has kids," Sharapova continued. "Talk about other things, but not draw attention to other things. She has so much in her life, many positives, and I think that's what it should be about."
Williams has been linked to coach Patrick Mouratoglou, but neither has confirmed their relationship extends beyond the court. When Mouratoglou was asked about the topic at the French Open this month, he smiled and replied: "Sorry. I don't understand the question."
According to the Rolling Stone story, posted online Tuesday, Williams spoke about what the reporter described as "a top-five player who is now in love."
Williams is quoted as saying: "She begins every interview with 'I'm so happy. I'm so lucky' it's so boring. She's still not going to be invited to the cool parties. And, hey, if she wants to be with the guy with a black heart, go for it."
That is followed by these words in parentheses from the author of the piece, Stephen Rodrick: "An educated guess is she's talking about Sharapova, who is now dating Grigor Dimitrov, one of Serena's rumored exes."
Sharapova beat Williams in the 2004 Wimbledon final. But Williams has won their past 13 matches in a row, including in the French Open final two weeks ago.
At Wimbledon, where play begins Monday, Williams is the defending champion and seeded No. 1. Sharapova is seeded No. 3. They only could face each other in the final.
Williams is scheduled to hold a pre-tournament news conference at Wimbledon on Sunday.
The Rolling Stone article, which was about 4,000 words, drew widespread attention mostly for a one-paragraph reference to the Steubenville rape case. Williams is quoted as saying the teenage victim "shouldn't have put herself in that position."
Two players from the Steubenville, Ohio, high school football team were convicted in March of raping a drunken 16-year-old girl; one of the boys was ordered to serve an additional year for photographing the girl naked. The case gained widespread attention in part because of the callousness with which other students used social media to gossip about it.
A day after the story was posted, Williams issued a statement in which she said she was "reaching out to the girl's family to let her know that I am deeply sorry for what was written."
Williams' statement continued: "What was written what I supposedly said is insensitive and hurtful, and I by no means would say or insinuate that she was at all to blame."
Said Sharapova on Saturday: "I was definitely sad to hear what she had to say about the whole case."