Bolsonaro, who has yet to concede to leftist incumbent Luiz Inacio Lula da Silva, left Brazil after refusing to comply with a Supreme Court order to surrender his passports.
The ruling relates to a money-laundering probe in which he denies wrongdoing.
The location of Flavio, his wife and two daughters was not immediately clear.
One source close to the family said Flavio had landed in Florida, but declined to say where.
"The State Department will issue visas to Flavio Bolsonaro and his family and allow them entry into the United States," a senior State Department official said on Tuesday.
Bolsonaro's departure threatens to deepen a rift between the two most populous nations in the Americas.
Lula has accused Washington of colluding with the Bolsonaros to put them back in power, and Reuters reported before Sunday's election on a US State Department campaign to restore them to office.
US President Donald Trump told reporters on Wednesday that he would approve of Flavio Bolsonaro coming to the United States.
"Well, I would approve. I think they're fine people. The election was very strange, and I thought it was terrible. It's gotten bad reviews. You look at what happened to their market", Trump said.
"I think it's so terrible to see it. I didn't take sides, you know. I know them both. I know Lula. I know Flavio a little bit. I know the father very well. The father was treated really badly. I thought, but I wasn't involved. That's Brazil. It's a different country", he said.
Trump has previously alleged Flavio's father, former president Jair Bolsonaro, was a victim of judicial persecution in Brazil.
The elder Bolsonaro also went to the United States after refusing to concede to Lula in 2022, when former US president Joe Biden was in office.
He eventually returned but was later given a 27-year sentence for plotting a coup after his supporters stormed government buildings in Brasilia.
Lula, speaking in Brasilia, said he had just heard the younger Bolsonaro had left the country, and called it shameful, saying he should have stayed in Brazil and proved his innocence.
Bolsonaro had been favoured to win Sunday's presidential election but ended up losing in a shock result.
On Monday, he was given 24 hours to surrender his passports, which were cancelled after the deadline lapsed, two sources said.
In a court ruling seen by Reuters, the Supreme Court cited the risk of Bolsonaro fleeing Brazil and his "significant financial resources" in ordering his passports seized.
Bolsonaro, who loses his immunity from prosecution when his Senate term ends in February, had said on Monday he had no plans to flee.
"I will hand over my passports and defend myself against this travesty, which is a clear attempt to intimidate me," he said.
His departure appeared to have caught his campaign manager off guard.
"We are surprised by the news because, just yesterday, we discussed the need for Flavio - given the 60 million votes he received - to take on this role as leader of the opposition," senator Rogerio Marinho told reporters.
Although Bolsonaro has yet to officially concede defeat - saying on Sunday he would seek the annulment of the result over alleged voter irregularities - his Liberal Party said on Monday it acknowledged Lula's victory.
with AP
