After a 16-day effort to free dozens of Indian construction workers trapped inside a Himalayan road tunnel, rescuers were finally preparing to pull the men out on Tuesday as diggers labored to clear a final stretch of debris by hand, the authorities said.

The rescue operation had hit repeated roadblocks, with officials ultimately trying multiple ways to reach the 41 stranded men. But a breakthrough came on Tuesday afternoon, as trained miners using hand tools made rapid progress after picking up at the point where a drilling machine had failed.

“The work of putting in the pipe to rescue the workers has been completed,” Pushkar Singh Dhami, the chief minister of the northern state of Uttarakhand, the site of the tunnel, said in a brief statement on social media. “Soon, all the worker brothers will be taken out.”

Syed Ata Hasnain, a member of India’s National Disaster Management Authority, gave a less definitive assessment and said that about two meters, or six feet, of drilling remained.

Speaking to reporters in New Delhi, he said at 4 p.m. local time, about two hours after the chief minister’s statement, that “we are near a breakthrough but not yet there.” The rescuers had moved close enough that the workers trapped inside could hear the preparations for their rescue, Mr. Hasnain said.

“In less than 24 hours, we have managed to dig 10 meters manually, which I would say is phenomenal,” he said. “There are 41 inside. Outside there are many more — the safety of those outside is as important as those inside, so we are not in a hurry.”

Once the rescue begins, he added, it will take about three or four hours to bring out all the workers through the inserted pipe, roughly three to five minutes for each one.

The workers’ ordeal, followed closely in India with regular updates on television and social media, put a spotlight on concerns long raised by environmental experts about large-scale construction in the fragile Himalayan mountain range. Experts say that the country’s procedures for environmental assessments of such projects are weak and prone to political interference.

The men were building a tunnel that is part of a major road project on a Hindu pilgrimage route when a landslide early on Nov. 12 trapped them behind about 60 meters, or about 195 feet, of debris.

Early on Tuesday afternoon, as officials reported that drilling had reached the final few feet separating the rescuers from the trapped workers, visuals from outside the tunnel showed a bevy of activity. Dozens of rescue workers in orange jumpsuits carried ropes and ladders, parked ambulances moved toward the tunnel, and prayers continued to be offered at a small makeshift roadside temple in the distance.

Family members of the stranded workers were told to be ready, as one relative would accompany each worker to the hospital.

“I will accompany Sanjay when he gets out. I feel at peace at the moment. We feel energized and happy to be told the ordeal will be over soon,” Jyotish Basumatary, the brother of Sanjay Basumatary, one of the trapped men, said by phone from outside the tunnel.

In the hours after the landslide on Nov. 12, officials were able to establish communication and confirm the workers were safe. A small pipe running into the tunnel was used to get them food, water and oxygen. About a week into their saga, an endoscopy camera sent through the pipe captured initial visuals of the workers, easing the concerns of their families.

But over the course of the two-week operation, assessments from officials that the rescuers would soon reach the workers had proved to be false alarms.

Initial drilling efforts were hampered by additional falling debris. And by Day 13, the rescue effort appeared in disarray as an American-made auger machine broke down with less than 20 meters to go in the drilling. As they tried to break up the auger and extricate it, officials initiated backup plans, including one in which workers began drilling vertically from the mountaintop.

New machines were flown in from different parts of the country. But, in the end, the rescue effort — aided by international tunneling experts — found success in manual drilling by “rathole miners” in the final stretch of the path that had been mostly cleared by the auger machine.

In India, rathole mining is a term for a method in which workers dig very small tunnels to reach coal.

Mr. Basumatary said he had talked to his brother eight or nine times since he became trapped. “The last time I spoke to him was last night. He said, ‘We are fine. We are getting food and clothes, mustard oil, chapatis, vegetable, lentils, rice and biscuits, apples and oranges.’”

Mr. Basumatary said that the workers had gone hungry the first day, but that basic food items — rice flakes, cashews and raisins — reached them the second day. Proper food, including hot meals, started reaching them about a week later, he said.

Most of the workers trapped in the tunnel were from India’s poorer states, such as Jharkhand, Odisha and Assam, places with high levels of migration for employment. Family members said they were working for salaries of about $250 a month.

“I am feeling very good — my heart, today, is tall like the mountain,” the father of one worker told television reporters outside the tunnel, pointing with his head to the mountain that had trapped his son.

The man, who gave his name as Chaudhary to reporters, said the government had helped him with accommodations as he waited for his son near the tunnel and had provided him the clothes he was wearing. The man had a backpack, and a television reporter asked him what he was carrying in it for his son, whom he would accompany to the hospital.

“Nothing. We have nothing, so what can I take for him?” the man said with a smile, as he unzipped the bag to show some clothes. “The clothes I am wearing were also given to me.”

“I will tell him, ‘Son, I am very happy today. The whole country, even the trees and plants, are happy,” he said.