Chairman Todd Boehly has previously said a multi-club model is a good pathway for young players to gain experience.
The size of the stake is unknown – however it has been reported the Blues will pay 75m euros (£65m) for close to 100% ownership.
Club president Marc Keller said it was an “important day” for Strasbourg.
The former West Ham midfielder added that while there was “no financial urgency” to the agreement, the shareholders were aware the club had “reached the ceiling of our model” and needed investment in order develop their ambition.
“This is an important day for Racing,” said Keller. “It’s something my shareholder friends and I have been thinking about for the past two years. We’ve built a club that’s healthy at every level and well managed.”
BlueCo, the consortium led by American Boehly that purchased Chelsea in a £4.25bn takeover with private equity firm Clearlake Capital in 2022, said: “It is an honour for us to be part of this historic club.
“We are committed to preserving the heritage of Racing and are focused on working closely with Marc and his management team to continue the excellent work they have been doing.
“This strategic investment would further our presence in European football, alongside our ownership of Chelsea. We believe it would create huge opportunities to share knowledge and expertise.”
Strasbourg said BlueCo intends to make an “active contribution” by investing in the men’s and women’s teams, plus the academy.
The French side would also be able to “exchange advice and expertise with Chelsea” and any other teams the owners are involved with.
The Blues have endured a difficult first season under their new ownership, sacking two managers and finishing in 12th place – their worst Premier League position for nearly 30 years.
Strasbourg also found themselves struggling during 2022-23 and finished 15th in the French top flight.