Sport says his recent performances have alerted the attention of various teams in Europe including Real Madrid.
Los Blancos are interested in Zaniolo but Roma chief Monchi has already moved quickly to tie down his emerging star.
As for Real Madrid, Zaniolo is another example of their attempts to bring in all the best talents from around the world.
The last was Brahim Diaz earlier this month. They've also recently signed Vinicius Jr and Rodrygo.