Rescuers said they believed the boys could stay alive by drinking fresh water in the cave, either dripping in through rocks or rushing in through the entrance.
Narongsak explained early Monday that fixing rope lines and deploying oxygen tanks along their route will allow the divers to operate.
"Any attempt to dive the boys and their coach out will not be taken lightly because there are significant technical challenges and risks to consider", The British Cave Rescue Council said.
"NAVY SEAL will come tomorrow, with food and doctors and everything", the rescuer said.
Relatives of the boys took shelter from heavy rain on Monday and were seen cheering, smiling and receiving calls after receiving the news.
The boys, aged 11 to 16, and their 25-year-old coach entered the sprawling Tham Luang Nang Non cave after a soccer game on June 23, but near-constant rains have thwarted the search for them. "You are very strong".
Two Thai medical workers have also joined the cave's occupants to keep them company and monitor their health, the SEALs added.
A huge global rescue effort has been under way, with teams slowly making their way through thick mud and high water to try to reach the group in a network that stretches 10km (6 miles) into a mountain.
The governor of Chiang Rai province, Narongsak Osatanakorn, said that all 13 were found safe at about 10:30 pm local time Monday.
"From chamber three to the intersection and then onto Pattaya Beach, this area is all flooded and dark", Apakorn said.
Meanwhile, other efforts to access the cave such as through drilling, and finding alternative passages, as well as draining water out of the cave, will continue.
Divers have thus far been thwarted by muddy water rising up in sections of the cave and forcing rescuers to withdraw over safety concerns, The Associated Press reports.
If diving proves impossible, there is an outside chance they can be drilled out or wait for waters to recede and walk out on foot.
Narongsak said officials had met and agreed on the need to "ensure 100 percent safety for the boys when we bring them out". Thailand's rainy season lasts until October, and the cave is regularly flooded.
BBC South East Asia correspondent Jonathan Head, who is at the scene, said the Thai military had a few doctors with the diving skills required to reach them.
Cochrane said thousands of soldiers have been scouring the mountain looking for ventilation shafts that could provide a back door to the cave. "The Royal Thai Government and the Thai people are grateful for this support and co-operation, and we all wish the team a safe and speedy recovery".
Emergency packages have been dropped into cave shafts containing food, beverages, a phone, a flashlight, candles, a lighter and a map of the cave, in the hope the boys will be able to find them.