Eight of the boys were brought out on stretchers over the first two days - four each on Sunday and Monday.
A medic and three Thai Navy Seals who have stayed with the boys on a small, dry shelf deep in the flooded cave will also come out, he said. They successfully extracted the second group of four a full two hours faster than the first, officials said.
Rescue leader Narongsak Osatanakorn said at a sudden press conference on Tuesday that the mission began at 10.08 a.m. local time and that everyone in the cave is expected to be rescued today.
Last Friday, an ex-Navy SEAL died while placing oxygen tanks along the 4.7-kilometer (2.9-mile) evacuation route.
British divers found the 13, huddled on a muddy bank in a partly flooded chamber several kilometres inside the complex, on Monday last week.
Health officials say the eight rescued so far are in good physical and mental health but more tests are being carried out.
Families of the first four boys were being kept at a distance because of fears of infection and the boys were initially only allowed to eat a rice-based porridge, authorities said.
Adisak Wongsukchan, the father of 14-year-old Akarat Wongsukchan who was rescued on Monday, said all he wants to do now is give his boy a hug.
Flash flooding trapped the group in the cave on June 23, sparking a multinational rescue mission that had to overcome numerous difficulties, including strong currents of murky water submerging sections of the cave.
"FIFA's priority remains the health of everyone involved in the operation and we will look into finding a new opportunity to invite the boys to a Federation Internationale de Football Association event to share with them a moment of communion and celebration".
However, another official, Thongchai Lertwilairatanapong, inspector general of the public health ministry, said initial blood tests "showed signs of infection".
The Navy SEALs announced the successful rescue result on Tuesday, completing an astonishing against-the-odds rescue mission that has captivated the world. Heavy rain has struck the region intermittently over the last three days and further downpours could set back draining efforts at the cave.
What happens when the players are out of the cave " The kids are footballers so they have high immune systems", Jesada said.
Miraculously, it looks like the Wild Boars team, with their current medical conditions unknown, may be able to make their invitation to the FIFA World Cup in Moscow on Sunday.
He wrote in a tweet posted this morning: "Mini-sub is ready if needed. Leaving here in case it may be useful in the future".
On Wednesday, Thailand's Prime Minister Prayuth Chan-ocha thanked people involved in the rescue.