The 61-year-old was reported to be driving, while fellow ex-Argentina and Tottenham team-mate Ricky Villa was a passenger.
Ardiles' son, Pablo, said his father had received treatment for head injuries but stated he is "fine" following the incident.
In quotes reported by BBC Sport, he said: "My understanding is he is fine.
"He has 20-odd stitches in his head and my understanding is he has been hospitalised overnight. But he's fine."
Ardiles is reportedly filming a documentary in the British territory, while Pablo's brother was also among the passengers.
He continued: "Ricky Villa and my brother were out there but they are OK and are back in the hotel.
"I don't know the exact details but I understand my dad was driving and that he lost control somehow.
"The main thing is he is fine, they are all OK."
Ardiles enjoyed a 10-year spell at White Hart Lane in which time he collected FA Cup and UEFA Cup winners medals, going on to manage the club between 1993-94.
A statement from the club's official Twitter account read: "All at (the) club wish Ossie Ardiles and Ricky Villa a speedy recovery after they were involved in a car accident in the Falklands last night."
