Broad, who was 38, was involved in the accident on Thursday.
He was working as part of former Chelsea boss Carlo Ancelotti's staff at Paris St Germain but before that he spent six years at Stamford Bridge until his departure in 2011.
Chelsea chief executive Ron Gourlay said: "This is tragic news. It is a tremendous shock to everyone at Chelsea Football Club. Our thoughts are with his family and everybody who knew Nick.
"He will be remembered at Chelsea as a popular and likeable member of staff. He was a dedicated and knowledgeable professional who worked tirelessly during a period when the club progressed behind the scenes and achieved great success on the pitch."
In a statement, Paris St Germain said: "It is with great sadness, and in accordance with his family, that Paris Saint-Germain announces the tragic death of Nick Broad following an automobile accident on Thursday.
"On behalf of Paris Saint-Germain, club president Nasser Al-Khelaifi offers his sincerest condolences to Nick's family.
"Nick, a long standing colleague of Carlo Ancelotti, had quickly created strong and intimate bonds with the rest of the club coaching staff, the players, the club executives, all employees and the entire Paris Saint-Germain family.
"All of us are stunned and devastated by the sudden death of a much-liked colleague, universally recognised for his professional expertise."