The 16-year-old is yet to make his first-team debut for Rayo but has trained with the club's senior side in the second half of the 2019-20 campaign.
Both Barca and Real Madrid have been strongly linked with the teenager, but according to Onda Cero, the reigning Spanish champions have won the race for his signature.
The report claims that Luzzi has penned a 'long-term contract' at Camp Nou, having turned down offers from a number of other clubs, with Real Madrid allegedly among those teams.
The striker's current deal with Rayo is due to expire at the end of June, meaning that Barca will only have to pay a small development rights fee in order to complete a deal.
Luzzi is expected to initially link up with the club's Juvenil A Under-19 side with the view to eventually progressing to Barcelona B.