Director Viktor Kossakovsky revealed the news at the Zurich Film Festival while promoting the new film Gunda, of which Phoenix served as executive producer.
Apologizing for the actor for not being able to be there, Mr Kossakovsky said: “He just had a baby, by the way his name was… a beautiful son called River so he can’t promote [the film] now. “
The baby is named after Phoenix’s older brother River, who died of a drug overdose at the age of 23.
Phoenix, 45, and Mara, 35, have not publicly confirmed the news.
This is their first child together after getting engaged in May of last year.
The couple met on the set of Her in 2013 and have since starred in several films together, including Mary Magdalene in 2018.
They never confirmed the pregnancy, but Mara was seen with a baby bump earlier this year.
River Phoenix died at the height of his career in 1993 while filming Dark Blood.
He was best known for his roles in Stand By Me, Indiana Jones And The Last Crusade, and My Own Private Idaho.
The late actor had performed at Hollywood club The Viper Room with friends including Johnny Depp and Red Hot Chili Peppers’ Flea when he overdosed and died in the early hours of October 31.
In a recent 60 Minutes interview, Joaquin opened up about his brother’s death, saying, “I feel like in pretty much every movie I’ve done there was a connection to River from a certain way.
“And I think we have all felt his presence and his guidance in our lives in many ways. “