Rosenior, 41, travelled to London to meet with Paul Winstanley and Laurence Stewart, with Chelsea looking to install a successor to Enzo Maresca, who left the London club on Thursday, as soon as possible.
The former Hull manager is the leading candidate to become Chelsea's fourth permanent boss since owners BlueCo, who are also in control of Strasbourg, took control of the Premier League side in 2022.
Rosenior will return to Strasbourg's Stade de la Meinau with an annoucement expected at a press conference there at 8.30am GMT on Tuesday.
A club source said Strasbourg are already "working behind the scenes to find his replacement... as quickly as possible".
Meanwhile Monday's training session was overseeen by Rosenior's assistants Jean-Marc Kuentz and Filipe Coelho.
Rosenior, who guided Strasbourg to a seventh-placed finish last season, was strongly tipped to take over at Stamford Bridge after Chelsea parted company with Italian coach Maresca on New Year's Day.
Chelsea were unhappy with reports that Maresca had spoken to Manchester City about possibly succeeding Pep Guardiola at the end of the season, amid suggestions that he had tried to use a potential move as a bargaining tool in negotiations over an extended contract.
There had also been a dispute over the extent of Maresca's authority to ignore the advice of medical staff when bringing players back into the team after injury.
Rosenior, however, will be expected to work under similar conditions, with issues such as player recruitment largely the responsibility of the club's five-strong team of sporting directors.
Chelsea - fifth in the Premier League three points off the Champions League places - have assembled a large squad of young players with limited top-level experience, with Maresca seemingly unhappy with the amount of squad rotation his side endured.
Rosenior has enjoyed a measure of success with a similarly young squad at Strasbourg, and led them into the knockout rounds of the Europa Conference League.
The club are currently seventh in Ligue 1.