Sexton has been out of action since suffering a hamstring injury during Ireland's defeat to England at the Aviva Stadium on Sunday, February 10.
Commenting on the Dubliner's recovery, team manager Michael Kearney said: "Johnny made good progress last week and over the weekend. As of now he has not been ruled out. We are happy with how he has gone but he is still at best a 50/50 call.
"He has only done running in straight lines. The plan tomorrow is that they will run him at as close to full speed as possible and take it on from there. At this stage it is a very difficult call to whether or not he will be available.
"He hasn't been ruled out but at best it is 50/50. He has done no kicking and he hasn't run anywhere near full out. Obviously tomorrow is going to be a very important day.
"If he can run full out tomorrow without having any adverse reaction, I think the plan would be that he would do some kicking Wednesday/Thursday. It (the decision) may be made tomorrow if he doesn't come through that full out running properly."
Second rows Donnacha Ryan (shoulder) and Mike McCarthy (knee) have made quicker progress in their respective recoveries and Kearney said the pair are both on course to be fit for Saturday's clash.
"Donnacha Ryan didn't train last week but progressed very well over the weekend. He took part in modified training this morning," he explained.
"He remains in some doubt for the weekend. Like any other injury, he will need to regain full function and train normally to be available...we're not as concerned as we would have been last week.
"He is an incredibly resilient character, Donnacha. He has been carrying bumps and bruises pretty well from the start of the Six Nations and his powers of recovery are very good, so I would be hopeful that he'd be okay."
Kearney added: "Mike McCarthy has done well. He trained fully on Friday and again this morning. Provided he suffers no adverse reaction he will be available for selection."
The remaining doubt surrounds young winger Craig Gilroy, Ireland's lone try last time out against Scotland. A groin strain forced him off at the hour mark in Edinburgh.
"Craig obviously came off during the game against Scotland and trained very lightly last week and this morning, so there would be some concern about him as well. He has a groin strain that is still troubling him a little bit."
The team announcement for the France game will take place at Carton House tomorrow afternoon (1.30pm). You can watch live on IrishRugby.ie.