Ossie Ardiles was recovering in a Falklands Island hospital on Tuesday morning after he was involved in a car accident which left him needing at least 20 stitches to his head.
Ardiles' former Spurs team-mate Ricky Villa was also involved in the incident, which took place while the pair were on the island to film a documentary.
Ardiles remains in the King Edward VII Memorial Hospital in Stanley, but his son Pablo said the 61-year-old was "fine".
Press Association Sport understands Villa has been discharged from the hospital.
"My understanding is he (Ardiles) is fine," Pablo Ardiles told the BBC.
"He has 20 odd stitches in his head and my understanding is he has been hospitalised overnight. But he's fine.
"Ricky Villa and my brother were out there but they are okay."
Ardiles and Villa spent 15 years on the playing staff at Tottenham between them.
A statement from the club read: "All at the club wish Ossie Ardiles and Ricky Villa speedy recoveries after they were involved in a car accident in the Falklands on Monday evening.
"We have been in touch with Ossie's friends and relatives and have been informed that both Ricky and he were fine, and that Ossie suffered a cut to the head that required stitches, but he remained conscious throughout. Ossie remains in hospital."
Pablo Ardiles told the Spurs website on Tuesday morning: "Thank you for all the messages of support from everywhere on Facebook and Twitter and people who have been messaging me and emailing me.
"Thank you very much to everyone supporting us. It's excellent to see people's reaction in a sense to what's happened, but thankfully everyone's fine."
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 to help the north London club to the UEFA Cup in 1984.
The Argentinian played 311 times for Tottenham. Ardiles also had a brief and unsuccessful spell as manager, but he is still revered by Spurs fans and is a regular at the club's games.
Villa joined Spurs at the same time as Ardiles and scored twice in the '81 FA Cup final replay win against Manchester City.
In the run up to the final, Tottenham released a song called "Ossie's Dream", which is still sung by Spurs fans today.
A spokesperson from the Falkland Islands Government later confirmed to Press Association Sport the details of the incident, with no further updates yet available from the hospital on Ardiles.
"I can confirm that there was a road traffic accident involving seven passengers last night in the Islands. Thankfully, due to the fast action of our rescue services and the British forces, they were successfully evacuated to hospital in Stanley, where their injuries were assessed," she said.
"Three members of the party remained in hospital overnight for observations and further treatment, and the remaining four were released from hospital following treatment for minor injuries."