"Congratulations @SerenaWilliams for your baby girl. I am so happy for you and I feel your emotion," Mouratoglou wrote.
"Btw ... I wish you a speedy recovery... we have a lot of work ahead of us," he added.
Williams and Ohanian have yet to officially confirm the birth, but the reports she had quickly became the talk of the tournament, which Williams has won six times.
"A baby girl? Well, I hope she doesn't play tennis," world number three Garbine Muguruza said with a smile during a news conference.
Fellow American Sloane Stephens said she expected to see a lot of the newborn.
"I am very happy for her," she said. "I cannot wait to see the little baby girl on tour because I am sure she will be around."
Williams confirmed her pregnancy in April hours after triggering frenzied speculation when she accidentally posted a short-lived selfie on social media with the caption: "20 weeks".
Williams, the world's highest-paid female athlete, was about two months pregnant when she captured her 23rd grand slam singles title at the Australian Open, one short of the all-time record held by Australian Margaret Court.
She told Vogue magazine last month about her "outrageous plan" to defend her title in Australia, where the year's first grand slam will be played from Jan. 15-28.
Other women have left the tour to have children and returned at a high level, although none has done so at Williams' age.
Kim Clijsters of Belgium retired and had a child before coming back at age 26 and winning three grand slam titles.
Australians Evonne Goolagong Cawley and Court also won grand slam titles after having children.
(Reporting by Frank Pingue in Toronto and Rory Carroll in New York, editing by Greg Stutchbury)
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