The miners have been stuck since Thursday in the wildcat Cabeza de Negro mine in southern Peru.
Peru's minister for women and vulnerable populations, Ana Jara, said Monday that, barring any drawbacks, the miners could be rescued by Tuesday night.
Her estimate is more optimistic than that of Carlos Bejarano, one of the mining engineers in charge of the rescue. He declined to give a specific estimate.
"A few hours ago, I had the satisfaction of being able to speak with our friends and peers who are trapped," he said. "They made some noises typical from miners to communicate in these cases and vice versa. We can't say how long it will take right now to get them out but I can guarantee that they are alive, that they are in good health, and that ultimately they are going to be freed alive."
A cave-in over the weekend complicated efforts.
"It's very complicated work. We're taking into account all the necessary security measures to avoid risks among the rescuers themselves," said Cesar Chonate, a regional head of Peru's civil defense agency, the state-run Andina news agency reported.
Video from state-run TV Peru showed workers, wearing hard hats and headlamps, loading rocks into a pushcart by hand.
It was not clear what caused the initial collapse.
The miners have been getting oxygen, food and water through a tube, which has also allowed them to stay in contact with people above ground, Andina reported.
Peruvian Mining Minister Jorge Merino was also in the area and appealed to mining companies for their expertise, according to a statement from his office.
Mining is big business in Peru, which is a major world producer of copper, silver, gold and other minerals.
"The important thing is that the nine people are alive. We won't abandon them," Merino said.
The ordeal stirred memories of the 2010 Chilean mine collapse, in which 33 men were trapped underground for 69 days. All of those miners were rescued, pulled one by one from hundreds of meters beneath the earth's surface with a specially designed capsule.
