Earlier on Wednesday, Bernard Kouchner, France's foreign minister, said: "Everything that we could have humanly done, we have done. Now we have to wait until our special envoys, the doctor, get to the area. We are expecting a lot from this."
There is no indication as to whether the Farc rebels have given the operation their blessing.
Referring to Kouchner's remarks, Luis Carlos Restrepo, Colombia's peace commissioner, said: "The mission is under way".
He said the Colombian government was complying with a request by France that it suspend all military operations against the Farc while the operation took place.
French media said a representative of the International Committee of the Red Cross (ICRC) would take part in the mission.
However, Barbara Hintermann, director of the ICRC in Colombia, said the medical mission was a French initiative that would not involve the Red Cross until the Farc asked it to participate.
Speaking in Bogota, she said: "We have had no contact with the Farc about this initiative."
Betancourt, 46, a former Colombian presidential candidate with dual French-Colombian nationality, has been held by the Farc for six years.
Speaking at a news conference in Paris earlier on Wednesday, Lorenzo Delloye-Betancourt, Betancourt's son, said these were the final hours of his mother's life unless critical measures were executed immediately
|Lorenzo Delloye-Betancourt said his mother|
was dying [AFP]
He said: "At the hour that I'm addressing you, in the Colombian jungle, a woman, my mother, is dying. She has Hepatitis B and leishmaniasis which requires a blood transfusion in the hours to come or risks losing her life."
He called on Manuel Marulanda, the Farc rebel leader, to allow his mother to receive medical treatment.
Delloye-Betancourt said: "To Marulanda, the world is watching you. It's up to you to decide whether to be remembered as a war criminal and be treated as one, or to enter the books as a human being.
"To President Uribe [Colombia's leader], I want to say that your strong initiatives in these last hours will re-generate our hope. But these initiatives should not be in any case, undertaken for mere symbolic purposes."
"You know that the plea I have just made will be my last call, there is no more time, either we will liberate my mother and the hostages, or we will bury her."
In an address to Marulanda on French televison on Tuesday, Sarkozy had called on the Farc leader to release Betancourt.
He said the Farc leader would be responsible for Betancourt's death and that failure to act now would close the doors to French and international support for a rebel peace deal.
Any successful mission to treat the captives would be the first contact for years with some of the hostages, whom the Farc rebels say they want to exchange for their jailed fighters.
Attempts to secure a deal to free the hostages, who also include three Americans, are deadlocked over a rebel demand that Uribe demilitarise an area in the south of Colombia for a safe haven to facilitate talks.
The Farc did release six captives earlier this year in a deal brokered by Hugo Chavez, Venezuela's president.