The Italian League (Lega Serie A) has confirmed the defender's transfer from the Bundesliga club has been completed, with the Italian side expected to announce it officially
Kjaer, 22, said he was frustrated with a hold up to the move on Tuesday after reports emerged there were doubts over a deal.
However, the transfer has now been confirmed by the Italian league, meaning an official announcement from Roma is likely to follow.
Kjaer thus returns to Serie A following a poor spell in Germany after leaving Palermo last year.
The Denmark international arrived in Italy last week for his medical before the move was finalised between the two clubs.