The 41-year-old has been the favourite to get the job at Stamford Bridge ever since Enzo Maresca departed on New Year's Day.
Rosenior has managed Strasbourg, owned by the same BlueCo investment vehicle as Chelsea, for the last 18 months.
He flew to London on Sunday for talks with the Chelsea hierarchy and at a press conference on Tuesday confirmed he will imminently be completing that move.
"I was fully focused on Strasbourg's last match but what has happened since is I was given permission to speak to one of the biggest clubs in the world," Rosenior said.
"It was an honour to be associated with a club like that and now it looks like I will be the next manager of that football club.
"I haven't signed yet but I have agreed it verbally with Chelsea.
“It is all agreed and will probably go through in the next few hours.”