The 41-year-old Englishman will replace Enzo Maresca, who left on New Year’s Day.
“I can’t turn down this opportunity to join an incredible club, an incredible team that are Club World Cup champions,” Rosenior said.
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Strasbourg is part of the multi-club ownership group BlueCo, owned by Todd Boehly and Clearlake Capital, which also controls Chelsea.
“I haven’t signed yet. Everything is agreed and will probably be finalized in the next few hours,” Rosenior said.
Rosenior said Strasbourg first-team coach Kalifa Cisse, assistant coach Justin Walker and head of analysis Ben Warner will join him at Chelsea.
Ronseior was appointed by Strasbourg in July 2024 and guided the French club to seventh place in Ligue 1 last season, securing European football for the first time in eight years.
Their last game was a 1-1 draw at Nice on Saturday, their fifth league game without a win.
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They are seventh in Ligue 1 and first in the Conference League.
“The last 18 months have been a joy and the best of my professional career,” Rosenior said.
“I am here because I care about this club (Strasbourg) and I felt it was right to answer your questions physically here today before moving forward.
“I will love this club for the rest of my life, but I can’t turn down Chelsea.”
Rosenior, a former Fulham and Hull City defender, has never managed in the Premier League.
“I wouldn’t have accepted the job at Chelsea if I wasn’t prepared,” he said.
“There are clubs you simply can’t turn down. I hope the Strasbourg fans can see that and be proud of it.
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“I have had interest from many clubs, including Champions League clubs, with whom I have always been open to (Strasbourg president) Marc (Keller) and our ownership.”
Chelsea are fifth in the Premier League and will face Fulham at Craven Cottage on Wednesday.
Interim manager Calum McFarlane took charge of Sunday’s 1-1 draw against Manchester City.
‘Rosenior wanted to look people in the eyes’ – analysis
It’s highly unusual for an outgoing coach to explain his departure at a news conference, but that’s exactly what Rosenior did Tuesday morning.
After flying to London to seal a verbal agreement with Chelsea to become their next manager, Rosenior returned to France and Strasbourg to say goodbye.
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At an extraordinary press conference at the Stade de la Meinau, the stadium he has called home for 18 months, he appeared emotional and allowed local journalists to ask difficult questions.
Rosenior spoke warmly about his favorite moments and said he wanted to look people in the eye and explain his reasons for leaving midseason.
Complicating matters is that both clubs are owned by BlueCo, which could become a flashpoint if this doesn’t go well.
But Rosenior seemed simply motivated to do the right thing and explain himself to his fans after a role that propelled him to what he called “one of the biggest clubs in the world.”