The 12 boys and their football coach were found alive on Monday and have received medical treatment and food.
Divers reached them nine days after they entered the caves in the north of the country and became trapped by rising waters caused by heavy rainfall.
More heavy rain could see water levels rise and threaten the air pocket where the group has taken refuge.
Attempts are being made to reduce the risk of this by pumping water from the underground system.
Professional divers are battling the difficult conditions to bring food and medicines to them, highlighting the complicated nature of any rescue.
A phone line is being installed so the boys can talk to their families.
Earlier, the Thai military said the boys would need to learn to dive – or wait months for flooding to recede before they could get out of the caves.
Food would need to be sent in for the next four months if the scuba-diving option was ruled out, the military added.
It is believed that most of the group cannot swim, complicating what would already be a difficult rescue.
The Thai authorities have appealed for donations of full-face scuba diving masks small enough to fit the boys in order to reduce the risk of their breathing apparatus coming loose as they travel through flooded passageways.
The group is also in a weak state after days without food.
Thai officials told reporters on Tuesday that seven divers, including a doctor and a nurse, were with the group inside the cave. They were providing health checks and treatment, and keeping the boys entertained.
“They have been fed with easy-to-digest, high-energy food with vitamins and minerals, under the supervision of a doctor,” Rear Admiral Apagorn Youkonggaew, head of the Thai navy’s special forces, told reporters.
“No need to worry. We will look after them as well as we can. We will bring them out safely.”
The admiral said the idea that the boys would have to remain in the cave system for four months was very much a worst-case scenario.
How were they found?
Two British rescuer divers who had flown over to join the search operation found the boys on Monday night.
The video of that first contact was posted on Facebook by Thai navy special forces.
The boys are seen by torchlight sitting on a ledge above water, responding to the divers that all 13 were there and that they were very hungry.
They ask how long they have been underground and whether they can leave now. The divers tell them they have to wait, but say people will come back for them.
One boy replies: “Oh. See you tomorrow.”
The search for the group had gripped the nation as it was unclear where they were or whether they were even still alive.
Families of the missing group were ecstatic at news of their discovery.
How did they get there?
The boys, aged between 11 and 16, and their 25-year-old coach went missing on 23 June. It is believed they entered the cave when it was dry and sudden heavy rains blocked the exit.
They were found on a rock shelf about 4km (2.5 miles) from the mouth of the cave.
It is thought the boys could move through parts of the cave in dry conditions but rushing waters clogged the narrow passages with mud and debris, blocking visibility and access.
One of the toughest stretches for the divers came as they neared the so-called Pattaya Beach – an elevated mound in the cave complex – where it was hoped the boys had sought refuge.
Divers had to navigate a series of sharp, narrow bends in near-darkness. They completed the difficult journey to find Pattaya Beach flooded, so swam on and found the boys about 400m away.
How can they get out?
Bringing the trapped boys to safety is an extremely dangerous task given the conditions inside.
The Tham Luang cave complex in Chiang Rai in northern Thailand is regularly flooded during the rainy season which lasts until September or October.
If the children are to be brought out before then, they will have to learn diving skills.
But experts have cautioned that taking inexperienced divers through the dangerous corridors of muddy, zero-visibility waters would be very risky.
Attempts to pump the water levels lower have so far not been successful.
If they are to wait until the water recedes by itself, it would mean the boys will have to stay in the cave for months and have to be continuously supplied with food and assistance.
Diver Ben Reymenants, who is assisting with the rescue mission, told the BBC that two Thai navy doctors had volunteered to stay with the boys for “up to four months” until the water recedes.
BBC South East Asia correspondent Jonathan Head, who is at the scene, says that if the option to bring them out is chosen, pulling the boys back through a long stretch of partly flooded tunnels will be a daunting challenge, particularly as the rainy season has just started and water levels will rise.
Other teams are still scouring the mountainside in the hope of finding another way into the cave.
Who are they?
The 12 boys are all members of a local football team and their coach is known to have taken them on occasional excursions and field trips.
Tinnakorn Boonpiem, whose 12-year-old son Mongkol is among the 13, told Agence France-Presse she was “so glad” to hear they were safe.
“I want him to be physically and mentally fit,” she said.
“I’m so happy I can’t put it into words,” another relative of one of the group told reporters as tears of joy streamed down his cheeks.