Roma have made a move for the Belgian after losing Nicolo Zaniolo for the season due to an ACL injury.
However, Mundo Deportivo says Roma's approach has sparked rival competition from across Europe with La Real indicating a willingness to do business.
While no names have been mentioned, one Premier League club and two Serie A rivals are also in contact.
Roma's last offer was to take Januzaj on-loan to June with an option to buy outright for €15m. However, La Real are insisting on a straight sale.