Satellite images have since revealed a path the kids took from the plane wreck, and rescuers have come across some of their belongings, a makeshift shelter and a half-eaten fruit.
Last week, they found a pair of shoes and a diaper.
"Based on the evidence, we concluded that the children are alive," rescue team leader General Pedro Sanchez told W Radio on Monday.
"If they were dead, it would be easy to find them because they would be still" and the sniffer dogs would find them, he added.
On the morning of May 1, a Cessna 206 airplane left a jungle area known as Araracuara heading for the town of San Jose del Guaviare in the Colombian Amazon.
Minutes after starting the 350-kilometer (217-mile) journey, the pilot reported problems with the engine and the plane disappeared from radars.
Between May 15 and 16, soldiers found the bodies of the three adults and the debris of the plane stuck vertically in the thick vegetation, its nose destroyed.
But the children -- Lesly (13), Soleiny (9), Tien Noriel (4) and baby Cristin were missing.
Some 200 soldiers and Indigenous people with knowledge of the terrain were combing a dense jungle area of some 320 square kilometers (124 square miles) -- about double the size of Washington, DC.
The air force had dumped 10,000 flyers into the forest with instructions in Spanish and the children's own Indigenous Huitoto language, telling them to stay put.
The leaflets also included survival tips, and the military has dropped food parcels and bottled water for the children.
Jaguars, pumas, snakes
On Sunday, the army placed powerful searchlights with a range of up to three kilometers in the area "so that the minors can approach us," search team member Colonel Fausto Avellaneda told the Noticias Caracol TV show.
Rescuers have also been broadcasting a message recorded by the children's grandmother, urging them to stay put so the soldiers can find them.
The general said the search team believed it had come to within 100 meters (328 feet) of the children, but storms, thick vegetation and marshy terrain prevented them from reaching the kids.
Air force helicopters and satellite images are being used in the search in an area home to jaguars, pumas, snakes and other predators, as well as armed groups that smuggle drugs and terrorize local populations.
With all hands on deck, members of the Indigenous community are holding traditional ceremonies "speaking to the jungle" and asking it to give up the children, according to the government.
The children are from the Indigenous Huitoto community, also spelled Witoto, known for living in harmony with the jungle.
Huitoto children learn hunting, fishing and gathering.
The kids' grandfather, Fidencio Valencia, has told AFP the children are well aquatinted with the jungle, but he feared "dark forces" were preventing their rescue.
On May 17, President Gustavo Petro said the children had been found alive, only to retract the announcement the following day, and apologize for the false information.