The group of teenagers and their 25-year-old coach are believed to have entered the cave on Saturday afternoon.
Rescue workers began searching for them on Saturday night after they were reported missing. They found bicycles along with sports equipment outside the cave.
Defence Minister General Prawit Wongsuwan said divers from the Royal Thai Navy reached the cave on Monday.
Later, local TV quoted the province’s deputy governor, Pasakorn Boonyalak, as saying he was confident the 13 were still alive after the divers found foot and hand prints on dry ground inside the cave in an area not reached by the water.
Rescue teams were preparing to train the team to use scuba gear, in case they had to dive out through areas of the cave which were completely submerged, he added.
Mr Pasakorn said helicopters had dropped food and water into a tunnel which residents of a village said gave access to the cave system, as well as handwritten notes instructing the group to wait for the rescuers, who would try to get to them there.