The singer and actor, 38, got fans excited when she introduced entrepreneur and Harvard University graduate Michael as her "fiance" when they met France's Prime Minister Gabriel Attal at the Paris Olympics on Sunday.
People has now reported Michael had proposed by the time the pop star celebrated her 38th birthday in March.
The outlet added Gaga did not wear her engagement ring to her celebratory dinner party at the time to avoid being photographed with it.
"The couple shared the happy news with family and close friends right away but wanted to keep the engagement private otherwise," a source said.
"Everyone's excited about them getting married … (Gaga is) the happiest with Michael."
The insider also told how Michael is "excited and supportive" of Gaga's career.
"He can't stand being in the spotlight but lets her shine."
They said he "has his own business priorities" which Gaga is "supportive of", and added the pair are "great together".
The source said they are also planning for more than a wedding as they would also "love to have a family".
Gaga - born Stefani Joanne Angelina Germanotta - told France's prime minister in Paris: "This is my fiance, Michael," before the politician shook Polansky's hand.
Attal went on to compliment Gaga's "stunning" and "breathtaking" performance during the Olympics opening ceremony on Friday.
Gaga and Michael first sparked engagement rumours in April when she was photographed wearing a massive diamond on her left ring finger.
The couple were first linked in 2019, and Michael has been at her side at a string of awards shows.
"My dogs and the man that I love are my whole life," Gaga told The Hollywood Reporter in 2021.
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