MercoPress - a South Atlantic news agency - on Tuesday morning said the 61-year-old had been involved in an incident while filming a documentary with former team-mate Ricky Villa.
The hospital in Stanley where Ardiles is reportedly being treated would not comment to Press Association Sport on whether or not he was under their care, but Tottenham did Tweet a message of support to him.
"All at club wish Ossie Ardiles and Ricky Villa speedy recovery after they were involved in a car accident in the Falklands last night," Spurs said.
BBC Sport also carried quotes from Ardiles' son, Pablo, in which he said his father was okay.
"My understanding is he is fine," he said. "He has 20 odd stitches in his head and my understanding is he has been hospitalised overnight. But he's fine."
Ardiles' other son is involved in the documentary and Pablo Ardiles added: "Ricky Villa and my brother were out there but they are okay. and are back in the hotel."
Ardiles signed for Tottenham in 1978 from Huracan, winning the FA Cup with them in 1981. He headed to Paris St Germain for a year on loan during the Falklands War, but returned for a further five years in 1983, winning the UEFA Cup in 1984.
He managed them in the 1993/94 season. Ardiles and Villa were inducted into Tottenham's Hall of Fame in 2008.