The midfielder spent a single season in English football, making 33 Premier League appearances and scoring four goals as Middlesbrough dropped back to the Championship.
De Roon started Boro's first Championship game of the season, a 1-0 loss at Wolverhampton Wanderers, but has been allowed to re-join his old club.
A £13 million deal would represent a small profit on the £12 million (A$19.8 million) Boro were said to have paid to sign the 26-year-old a year ago.
Atalanta finished fourth in Serie A under Gian Piero Gasperini last season - above Inter Milan, AC Milan and Lazio - to qualify for the group stage of the UEFA Europa League.
"During my year away from Atalanta, I kept in touch with so many players but also with the staff," De Roon told reporters when he arrived in Italy to finalise his transfer.
"They have played an incredible season, the best ever, and what they have achieved last season was crazy."