Pulisic, 20, agreed to a reported €64million move from Borussia Dortmund in January before finishing the season on loan at the Bundesliga club.
The United States international may get a chance to play alongside Hazard, although the Belgian star is linked with a move away from Stamford Bridge.
However, Pulisic said he wanted to match Hazard, who has helped Chelsea win two Premier League titles, an FA Cup, an EFL Cup and a Europa League.
"It is incredible to see what Eden can do. He is a guy to look up to and what I would love to become," Pulisic said. "It is definitely a goal. Any player would be dumb not to want to be in the same team as him."
Chelsea, which faces Arsenal in the Europa League final on 29 May, finished third in the Premier League, but it lagged 25 points behind second-placed Liverpool.
Pulisic believes his team can close the gap to Liverpool, and champion Manchester City.
"We are a confident group of guys," he said. "Liverpool and City are two great teams who had great seasons but I have seen Chelsea compete against big teams and do well against them this season.
"We want to go in with a winning mentality and compete with them right away."