SEAL commander rear admiral Arpakorn Yookongkaew said seven members of his unit - including a doctor and a nurse - are now with the group.
Rescuers are racing to pump water from the Tham Luang Nang Non Cave before the predicted arrival of heavy rain, which would complicate efforts to free the team, who have been trapped 1km underground for the last 10 days.
But it is thought the boys, who cannot swim, will either have to learn to swim and then to scuba dive through very narrow passages, or face waiting as long as four months for waters to subside.
Mr Yookongkaew told a news conference his team members "have given the boys food, starting from easily digested and high-powered food with enough minerals".
He said that having the boys dive out of the cave was one of several options being considered. If it were employed, he said they "have to be certain that it will work and have to have a drill to make that it's 100 percent safe".
In video shot by British divers as the group were found, torchlight reveals boys in shorts and red and blue shirts in an underground cavern beside an expanse of water.
Thailand cave rescue operation: in pictures
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With the onset of the rainy season, Thai navy captain Anand Surawan had initially warned the group could be trapped for months.
"[We will] prepare to send additional food to be sustained for at least four months and train all 13 to dive while continuing to drain the water,” he said.
Police have said the 25-year-old football coach who led the team into the cave network could face legal action.
An international team of divers, including from Britain, struggled through narrow passages and murky waters to find the boys, aged between 11-16, and their coach late on Monday night on an elevated rock about 2.5 miles from the mouth of the cave.
Care packages dropped for boys missing in Thailand cave
"How many of you are there - 13? Brilliant," a member of the multinational rescue team, speaking in English, tells the boys. "You have been here 10 days. You are very strong."
News of the boys' survival sparked jubilation in a nation that has been gripped by the harrowing drama.
"Thank you," one of the boys says.
One boy asks when they will get out, to which the rescuer answers: "Not today. You have to dive."
The boys were found in weak condition, but with only minor injuries.
Interior minister Anupong Paochinda said rescuers needed to intensify efforts to reduce water levels in the cave.
Rain continued to fall in Chiang Rai on Tuesday and was forecast to intensify from Wednesday.
The Thai government has released advice on the rehabilitation of the football team if and when they get out.
It says survivors and family members may experience post traumatic stress disorder (PTSD). Once out, they should be provided with a "calm and peaceful environment" to create "a feeling of safety and stability".
"Families should also consult with a psychiatrist should the children exhibit symptoms of PTSD, so that they can receive the necessary therapy or medication, and be available and attentive if the child wishes to discuss their feelings or experiences," it adds. "Families should also encourage their children to engage in stress-free activities and avoid leaving the children alone."
Thai interior minister Anupong Paochinda has said the boys will be brought out the same complicated route the rescue diverse took to find them.
He said efforts to pump out the water would continue, but that it was clear there were areas which could not be drained.
The children might need diving gear, he added, and would be accompanied by two divers each on their way out, according to the Bangkok Post.
''Diving is not easy. For people who have never done it, it will be difficult, unlike diving in a swimming pool, because the cave's features have small channels,'' he said. ''If something happens midway, it could be life-threatening.''
The young footballers went into the cave as part of an initiation, one of the divers who helped locate them has said.
The boys reportedly went into the Luang Nang Non Cave in order to write their names on the walls.
They were "wading in and trying to go to the end of the tunnel, sort of like an initiation for local young boys to… write your name on the wall and make it back," Ben Reymenants told Sky News.
The Thai government has published advice on how the boys should be rehabilitated once out of the caves. It asks the public to be considerate of what they publish on social media, and that the boys should not be exposed to images of the incident to avoid psychological repercussions.
Child psychiatrist Dr Tirapon Tungchittipon said they may suffer from post traumatic stress disorder (PTSD).
“Survivors of disasters or traumatic events will exhibit varying coping mechanisms depending on the cause of the event and pressures involved. Environmental factors, such as family background, social support, coverage of the event, as well as the opinions of others, all have an impact on a child’s mental well-being,” the guidance says.
Rick Stanton and John Volanthen were the first to reach the 12 boys and their coach inside the Luang Nang Non Cave in Chiang Rai province on Monday.
It is their voices that can be heard talking to the boys and giving them calm reassurance in a dramatic video released by the Thai navy. They are working with Thai navy SEALs, who are leading the rescue operation.
Stanton, the retired firefighter from Coventry, and Volanthen, who does IT consulting work in Bristol, have years of experience in cave rescues and have helped map the Luang Nang Non Cave.
"Brits are probably one of the best cave-diving teams," said Dinko Novosel, head of the European Cave Rescue Commission, even before the search mission proved successful.
Gary Mitchell, the assistant vice chairman of the British Cave Rescue Council told AP: "They may start to dive them out in small batches into small, into other chambers, other pockets of air, and do that over a staggered period of time."
"Obviously, the biggest risk really is current rainfall. If it keeps raining or starts to rain and water levels rise then there's a bit more immediacy required."
Drilling an opening would be a problem because of the need for determining with pinpoint accuracy where they are inside the cave. While stressing that the Thai military is in charge, Mitchell believes they are working under the assumption that the group is about 800 meters to a kilometre below the surface — meaning the need to drill through solid rock.
"Drilling from the top of a jungle mountain ridge is really tricky," he said.
Thai navy SEALs say all 13 people trapped in a flooded cave in northern Thailand are healthy and being looked after by medics.
SEAL commander Rear Adm. Arpakorn Yookongkaew said that seven members of his unit — including a doctor and a nurse — are now with the 12 boys and their coach in the cave where they took shelter.
He told a news conference that his team members "have given the boys food, starting from easily digested and high-powered food with enough minerals."
He said that having the rescued people dive out of the cave was one of several options being considered. If it were employed, he said they "have to be certain that it will work and have to have a drill to make that it's 100 per cent safe."
He said there was no rush to bring them out as they are safe where they are.
A miner who spent 69 days trapped underground in Chile says the 12 boys and their football coach stranded in a cave in Thailand should help each other mentally prepare for rescue.
Omar Reygadas and 32 other miners were rescued in 2010. He says the ordeal was hard for grown men so he can only imagine what the boys, aged 11 to 16 are going through.
Mr Reygadas said their coach will be a key figure in keeping them motivated. Beyond that they should focus on being reunited with their families.
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