"[Pulisic] can be one of the best in the future for sure," Hazard told The Associated Press at his unveiling as one of the covermen for EA Sports' FIFA 20 video game in New York.
Hazard made his long-awaited move to Real Madrid this summer after months of indication that his days at Chelsea were numbered. The Spanish giants are coming off one of their worst seasons in club history, finishing the campaign without a major trophy.
Hazard became Madrid's first major signing since Cristiano Ronaldo left for Juventus last summer, a move that cost Real about $113 million in bring in the Belgium international.
Hazard thinks Pulisic can follow in his footsteps at Chelsea.
"He can be [a star]," he said of Pulisic. "Now he's at one of the best clubs in the world...He's a proper player."
Chelsea signed Pulisic from Borussia Dortmund for $73.1 million -- a record deal for an American -- and loaned him back to the Bundesliga club ahead of him moving to the Premier League in the summer.
Pulisic made his debut for new boss Frank Lampard in Chelsea's friendly win over Barcelona earlier this week in Japan, saying "it's been tough getting into the things but I'm feeling really good."
Hazard and Real Madrid will face city rivals Atletico Madrid in a International Champions Cup match on Friday at MetLife Stadium in East Rutherford, New Jersey.
The Associated Press contributed to this report.