According to Globo Esporte, the 23-year-old is signing a five-year deal with the principality club.
Monaco will pay €8m initially with a further €2m in variables.
Benfica and Porto also had Henrique in their sights, but negotiations did not progress very far.
Porto were keen on him to replace Alex Telles, but Telles’ proposed move to PSG hasn’t taken place.
Henrique joined Atletico’s youth setup for Santos in 2016 but has spent the last three years back in Brazil on several loan deals.
Atleti used a clause to cut short Henrique’s latest loan at Gremio and Diego Simeone planned to use him as a backup to Renan Lodi.
However, the Brazil under-23 international wanted to move on to get more chances and his wish has been granted.
Henrique will now join up with Monaco’s ambitious project under the tutelage of Niko Kovac.