The rent-controlled apartment where Bernard Dethiers has lived for more than 40 years is his lifeline to staying in San Francisco.

Mr. Dethiers, 74, is retired and pays around $1,100 a month for his two-bedroom in North Beach, significantly below the city’s median asking rent in August, $4,395. His son, a senior in high school, splits his time between the apartment and his mother’s home nearby.

But Mr. Dethiers is at risk of losing it. After new owners bought the building last year, they offered him $40,000 to leave within about three months. He countered for $600,000, a number he reached by adding up the cost of a “potential rent differential” of $2,500 a month over 20 years. The owners rejected it.

“We need to be in San Francisco, and we have nowhere to go because of the price,” Mr. Dethiers said. “We are totally priced out.”

Now the owners are seeking to evict him and five other households under the Ellis Act, a California state law that allows landlords to evict tenants in order to remove all of the property’s rental units from the market. (The owners of the building intend to live in the units with their families, said Gael Bizel-Bizellot, a lawyer representing them.) Mr. Dethiers and the other tenants are fighting the eviction in court.

As the artificial intelligence boom sends rents and home prices soaring in San Francisco, landlords are initiating buyout negotiations, presenting many tenants with a dilemma: Accept a buyout offer or risk no-fault eviction, in which they would receive only the relocation payment required by law.

Before beginning buyout negotiations, landlords must disclose their intention to tenants and the San Francisco Rent Board. According to Rent Board data, the number of pre-buyout declarations filed by landlords rose about 17 percent in the first half of this year from the same period last year. So far this year, Ellis Act and owner move-in notices together made up about 10 percent of eviction notices filed with the Rent Board.

“The relocation money simply does not reflect what the tenants are giving up,” said Daniel Wayne, a lawyer who represents tenants in San Francisco. For people who want to remain in the Bay Area, Mr. Wayne said, current rent prices and moving costs can quickly eat up the money from a buyout or a relocation payment resulting from a no-fault eviction.

Mr. Dethiers and his son are entitled to about $29,000 in relocation payment. Because he is over 62 and has lived in his apartment for more than a year, they would have a year to move instead of the typical 120 days. With the city’s high rents, Mr. Dethiers said he doesn’t know where in San Francisco he would live if he and the other tenants lose their case.

“I cannot think of anything worse, really, than losing your place and knowing that you’re not going to be able to get another one,” he said.

Owners may pursue buyouts if they want to charge market-rate rent to a new tenant or sell a building that could be more valuable when vacant, said Scott Freedman, a lawyer for property owners in San Francisco. Some of the owners and landlords he represents moved away but now want to return to their homes in San Francisco. Others have children who want to move in after college.

“It’s similar to the tech booms in the past when people got really excited about what was going on around here and they wanted to be a part of it,” Mr. Freedman said. “There’s a push in demand. There’s only so much supply.”

But rent control gives many tenants leverage in negotiations since annual rent increases are generally limited, and they can choose to reject an offer or refuse to negotiate. Tenants also have 45 days to cancel a signed buyout agreement. If a buyout agreement isn’t reached, landlords can evict tenants under the Ellis Act by removing all of the property’s rental units from the market. An owner may also pursue a no-fault eviction so they or certain family members can use the unit as their primary residence, but they must live in the unit for at least three years.

Craig Lipton, 59, has worked in real estate in San Francisco for about 35 years and wanted to take advantage of the competitive housing market driven by A.I. wealth. He planned to sell part of his luxury two-unit building in Nob Hill, and in March, he started negotiating with five tenants who split a rent-controlled five-bedroom unit for $11,000 in monthly rent. (Mr. Lipton, who lived in the other unit, said he wanted to buy out the tenants because he didn’t think he would be able to sell the unit with them in it.)

Mr. Lipton first offered the tenants $5,000 if they left by the end of June. They declined. He then offered a buyout worth $55,000, and they countered with one worth $152,000. They agreed to a buyout worth $100,000.

“Having to pay to secure possession back of my home did feel a little bit like I was being held hostage,” Mr. Lipton wrote in a text message.

At the end of July, Mr. Lipton listed the entire building for $7.25 million.

John Alioto, 47, was born and raised in San Francisco and has lived in his rent-controlled apartment in Nob Hill for more than 20 years. As a doorman at the Hyatt Regency, welcoming people to San Francisco is the essence of his job. Now, after his landlord offered to pay Mr. Alioto and his housemates to leave, Mr. Alioto worries he may have to say goodbye to the city.

Mr. Alioto and his three housemates split about $2,600 a month for a four-bedroom apartment in a three-unit building. Their landlords, Shivashis Nayak and Jeetendra Pradhan, offered Mr. Alioto and his three roommates $70,000 in April to move out within about two months. After they didn’t accept, the landlords raised the offer to $104,000. The tenants did not accept that offer, either.

“I don’t think you could put a price tag on leaving your home,” Mr. Alioto said. “There’s no amount of money that could be offered in a briefcase outside of our house for us to say, ‘Oh, this is great. Let’s go.’”

In his ideal world, Mr. Alioto would reject future buyout offers to keep living in his apartment, but he worries about the possibility of a no-fault eviction.

“The ultimate fear is just not having anywhere to live that’s comfortable that you could call home,” Mr. Alioto said. “Where am I gonna end up?”

A lawyer for Mr. Nayak and Mr. Pradhan declined to comment.

As demand for housing remains strong, Mr. Freedman expects an uptick in buyout negotiations.

Mike Dineen, 53, a native San Franciscan who works at the worker-owned Rainbow Grocery Cooperative, has lived in the unit above Mr. Alioto’s for nearly 30 years and splits about $2,600 in monthly rent with three housemates. They also received buyout offers.

Mr. Dineen did not respond because he considered the offers too low.

“You get nervous, and then you start doing your homework,” Mr. Dineen said of recognizing his rights. “The more you learn, you feel a little safer, but you’re still in limbo. So there is no peace.”

His apartment is where he moved his mother in during the last year and a half of her life, where he became her full-time caretaker and where he met neighbors who, over decades, became his greatest source of community, he said.

The idea of not finding an affordable apartment and having to leave San Francisco and find a new job in the current market is frightening, Mr. Dineen said.

“I’m not getting any younger,” Mr. Dineen said. “I’m slowing down, not ramping up.”

Mr. Dineen and Mr. Alioto have lived in the building longer than they’ve lived anywhere else.

“I don’t own the building,” Mr. Dineen said, “but this is my home.”