The four-times Tour champion sustained multiple fractures including broken femur, elbow and rib bones when he crashed at high speed while checking the course ahead of the individual time trial on Wednesday.
"Chris woke up this morning and was reviewed by intensive care consultants and the orthopaedic specialist who operated on him and they're both very happy with his progress," team doctor Richard Usher said in a statement.
"He will remain in hospital for the next few days for observation, but he's already actively engaging in discussing his rehabilitation options, which is very encouraging.
"As he begins his road to recovery, the team will now only be providing further updates at a stage where it is necessary."
The team said that Froome had offered his thanks for the overwhelming number of messages of support from fans and would release a statement of his own in the coming days.
In Froome's absence, defending champion Geraint Thomas and Colombian Egan Bernal will lead Team Ineos on the Tour, which starts in Brussels on July 6, although the team tactics have yet to be decided on.
"We have a team with great riders. Everybody can be in top shape in July. As soon as we are at the start of the Tour we will make decisions," team manager Dave Brailsford was quoted as saying by local newspaper Le Dauphine Libere.
"You don't set up tactics for July in June."
The British outfit have won six of the last seven Tours as Team Sky before being re-launched as Team Ineos earlier this year.
(Reporting by Julien Pretot and Shrivathsa Sridhar; Editing by Christian Radnedge)
