Reuters could not confirm the identity of the three people brought out in the evening and the chief of the rescue mission, Narongsak Osottanakorn, declined to comment, saying a news conference would be held later on Monday.
The "Wild Boars" soccer team and their coach got trapped on June 23 when they set out to explore the vast cave complex after soccer practice, when a rainy season downpour flooded the tunnels.
British divers found the 13, huddled on a muddy bank in a partly flooded chamber several kilometres inside the complex, on Monday last week.
The dangerous bid to rescue the boys - aged between 11 and 16 - got going again hours earlier on Monday after a break to replenish oxygen supplies and make other preparations deep inside the cave complex.
Authorities have said the mission could take three or four days to complete. It is a race against the clock with heavy rain expected in coming days, which would again dangerously flood the tunnels with fast flowing, and rising, water.
The rescue team went into the cave to resume the operation at 11 a.m. (0400 GMT), Narongsak told a news conference earlier, adding he expected good news.
Thirteen foreign divers and five members of Thailand's elite navy SEAL unit make up the main team guiding the boys to safety through narrow, submerged passageways that claimed the life of a former Thai navy diver on Friday.
Narongsak said that the "same multinational team" that went into the cave on Sunday to retrieve the first four boys was deployed on Monday.
He did not say how many boys the team hoped to bring out on Monday.
On Sunday, divers held the first four boys close to bring them out, and each had to wear an oxygen mask to enable normal breathing, authorities said.
Narongsak said rescuers had to tighten a guide rope as part of their preparations for the second phase of the rescue on Monday.
Interior Minister Anupong Paojinda told reporters the four boys rescued on Sunday were in good health in hospital but did not give details. There was no word on the condition of any of the people brought out on Monday.