The Blues are ready to pay £45m to beat off competition from Liverpool and Arsenal for the American international.
The 20-year-old playmaker, who has been a Chelsea target since the start of the season, is set move to Stamford Bridge in the summer.
Pulisic played as a seven year-old in the youth team of National League North Brackley Town in Northamptonshire when his teacher mother Kelley was on a work exchange.
He only has one more year left on his contract with Borussia Dortmund, where he was brought through from the youth team by Jurgen Klopp.
The Blues will only consider parting with Hazard who is still stalling on signing a new deal once they have got a replacement signed and sealed.
Real are ready to offer the Blues around £150m for the Belgium international as well as at least one player.
They are ready to let Mateo Kovacic make his loan move permanent as part of the deal, while Isco could also be added the mega package to get their man.
Chelsea boss Maurizio Sarri issued Hazard an ultimatum and called the lingering transfer rumours surrounding the winger ‘a problem’.
“I think we have to solve the problem,” he said. “If we want to plan the future, we have to solve this problem.
“I don’t know [what Hazard is thinking], but I think that it’s time to decide.”
Hazard will enter into the final year of his contract at the end of the current season but recently said he might be happy to sign new terms.
He became just the tenth player to reach the milestone of 100 goals for the club in the recent 2-1 win over Watford and gave fans some cause for optimism with his comments after the game.
”It means a lot, especially because we also won,” Hazard said.
“Scoring 101 goals with this amazing club is something I will never forget.
“But now I think the fans, the staff, myself, the players, we want more.
“I want to score more for this club and then try to be a legend like [Frank] Lampard, [John] Terry, [Didier] Drogba, I try to do my best.”
Although Hazard is unsure on where his future lies, in October he admitted that he has always dreamed of playing for Real Madrid.
He said: “I am playing good football. Real Madrid is the best club in the world. I don’t want to lie today.
“It is my dream since I was a kid. I was dreaming about this club. I don’t want to talk about this every day.
“I don’t have time, but we will talk about my future soon. I think so.”
Hazard has already scored 10 Premier League goals this season and assisted a further nine.
Chelsea face Crystal Palace on Sunday with the Blues looking to secure their grasp on a top four spot.