World number two Naomi Osaka faces expulsion from the French Open if she continues to refuse to speak to the media, Grand Slam organisers said.
Japan's Osaka said last week she will not give any news conferences during the Grand Slam because she wants to protect her mental health.
She was fined $15,000 (£10,570) for not doing media after her first-round win against Romania's Patricia Maria Tig on Sunday.
The second seed won 6-4 7-6 (7-4).
A joint statement from the four Grand Slam organisers said Osaka also faces "more substantial fines and future Grand Slam suspensions".
They added that following her announcement, Roland Garros teams asked the 23-year-old to reconsider her position and were unsuccessful in their attempts to check on her wellbeing.
After a "lack of engagement" from Osaka, the Grand Slam organisers wrote to her to offer support, as well as to "remind her of her obligations".
"A core element of the Grand Slam regulations is the responsibility of the players to engage with the media, whatever the result of their match, a responsibility which players take for the benefit of the sport, the fans and for themselves," the statement said.
"We have advised Naomi Osaka that should she continue to ignore her media obligations during the tournament, she would be exposing herself to possible further Code of Conduct infringement consequences.
"As might be expected, repeat violations attract tougher sanctions, including default from the tournament and the trigger of a major offence investigation that could lead to more substantial fines and future Grand Slam suspensions."
It added that the rules exist "to ensure all players are treated exactly the same".
Osaka had earlier moved to put the controversy to one side as she started her campaign in Paris with a straight-set win over Tig.
She hit 20 winners in a largely dominant first set on Court Philippe Chatrier, but had to work even harder in the second before her quality came through in the tie-break.
The four-time major champion plays another Romanian, Ana Bogdan, next.
Osaka has not yet reached the same heights on the red dirt that she has on hard courts, having never progressed past the third round at Roland Garros and holding a win-loss record of 13-13 in the events leading up to the Grand Slam.
How did it get to this point?
On Wednesday, Osaka released a statement saying she would not face the media during the course of the French Open, citing the impact of news conferences on the mental health of players.
She said expecting players to answer questions after a defeat amounted to "kicking a person while they're down".
"I've often felt that people have no regard for athletes' mental health and this rings true whenever I see a press conference or partake in one," she said in the statement, which she posted on social media.
"We're often sat there and asked questions that we've been asked multiple times before or asked questions that bring doubt into our minds and I'm just not going to subject myself to people that doubt me."
Grand Slam rules state players can be fined up to $20,000 (£14,160) for failing to meet their media obligations, with the Women's Tennis Association (WTA) saying the players "have a responsibility to their sport and their fans" to speak to the media during competitions.
Many athletes, both in and out of tennis, applauded Osaka's stance, though many accepted that speaking to the media is "part of the job".
Novak Djokovic, the men's world number one, said: "I understand the press conferences sometimes can be very unpleasant, and it's not something you enjoy always - especially if you lose a match.
"But it is part of the sport and part of your life on the tour, and this is something we will have to do."
Iga Swiatek, the reigning French Open women's champion, added that while media is "not the most enjoyable thing to do", it is "important".
Following her win over Tig, Osaka did take part in the usual on-court interview with the victor.
"My movement on clay is a work in progress. If I keep playing more matches, hopefully it will get better," she said.