Nicolo Zaniolo has started to be rumoured with Tottenham this time. I saw not much time ago that Juventus had an interest to sign him this summer. But now rather than Juventus, Tottenham are likely to sign him. It is still bad news for Roma of course as they wouldn't like to lose one of their most important players.
It is still a rumor, AS Roma doesn't need to worry.
As far as I know, Zaniolo was ever rumored to join AC Milan but he stated that he is happy to be AS Roma player. Till now, there is no progress on the rumor, Zaniolo still stays in AS Roma. So, I believe the current rumors that he probably joins Juventus or Tottenham, will end the same as the previous rumor. Since Zaniolo is the key player in AS Roma squad, he must prefer to stay.