But, their conditions are unknown.
A source at the Chiang Rai hospital said that five emergency response doctors were awaiting the boys and a further 30 doctors were on stand-by.
He says the divers went in at 10 a.m. and the boys will gradually come out accompanied by two divers each.
"It's risky to the most experienced divers to go through", said one diver who spoke to Reuters.
Two Britons, John Volanthen and Richard Stanton, are understood to have a leadership role and were among the divers who found the boys and their coach marooned on a bank inside the cave.
After an initial assessment at the site, the boys were airlifted to a makeshift helipad close to the Chiang Rai Prachanukroh hospital, some 70km away.
Narongsak was then seen shaking hands with a senior army figure and watching intently at the entrance of a long green medical field tent beside around 15 soldiers.
Mild weather and falling water levels over the past few days had created optimal conditions for an underwater evacuation that won't last if it rains again, Narongsak had said.
A treacherous rescue bid to free a youth football team trapped in a flooded Thai cave entered its second day Monday, with nine of the "Wild Boars" still inside after elite divers guided four out.
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"The kids are so strong, physically and mentally", Narongsak said. Once the team are ready, they will do so immediately.
Jake Zweig said on "Fox & Friends" that he was confident in the SEALs working to rescue the 12 boys and their coach, but that there is a possibility of a rescuer losing his life.
"Narongsak Osatanakorn, the head of the joint command centre coordinating the rescue, said Sunday morning was the most opportune time to launch the operation, citing the ideal weather, water and the kids' readiness, physical and mental".
"If something changes, we'll stop", he said. "But I expect the operation to finish in the next couple of days".
Dozens of divers have arrived at the cave on Sunday morning.
Thailand's Meteorological Department said there was a 60 percent chance of rain Monday with thunderstorms forecast throughout the week. The boys sounded calm and reassuring in handwritten notes to their families that were made public Saturday. One of the boys, identified as Tun, wrote: "Mom and Dad, please don't worry, I am fine".
Expert climbers, divers and Thai Navy Seals have mulled contingencies ranging from drilling an escape route through the mountain to waiting out the monsoon inside the cave.
The boys wore "full face masks and the rescue divers carried them out through the passage in the cave complex", Governor Osotthanakorn told a news conference after the rescue.
"There's no time limit on the operation", he said.