A royal source told The Sun: "Andrew has made it quite clear that he wasn't leaving Balmoral. It would appear the advice to him was to stay in hiding and miss the whole thing."
A lawsuit was brought against Prince Andrew by Virginia Giuffre, which alleges she was coerced into having sex with him at age 17. Andrew has repeatedly denied the allegations.
It is believed Andrew will remain at Balmoral for three weeks amid the legal issues.
Princess Beatrice welcomes baby girl
Beatrice is yet to reveal the name of the 2.7kg baby girl.
A statement from Princess Beatrice and her husband Edorardo Mapelli Mozzi said they were "delighted". The couple wed in July 2020.
"The new baby's grandparents and great-grandparents have all been informed and are delighted with the news. The family would like to thank all the staff at the hospital for their wonderful care.
"Her Royal Highness and her child are both doing well, and the couple are looking forward to introducing their daughter to her big brother Christopher Woolf."
The baby girl is 11th in line to the throne.