A CENTURY ago, an Italian bricklayer named Donato Salvucci had a problem: He couldn’t get into the union, so no one would give him a job. A friend suggested he change his name to something that sounded Irish. After he became “Dan Sullivan’’ he got hired right away.

The history of a city is, in many ways, the story of the unions that built it. Boston’s is a Homeric saga of one immigrant group after another struggling to become members of that blue-collar club. All too often, once they gain acceptance, union members favor their own. Apprenticeships have often gone to relatives, friends, or neighbors. And who could blame them? A union job is a ticket to a middle-class life. Nobody hands those out to strangers.

But today, a new crop of union leaders - including Dorchester State Representative Marty Walsh - are pushing to change the way unions do business. Walsh, secretary-treasurer of the Boston Building Trades Council, is egging unions on to recruit those who have traditionally been left out.

And he should be. Expanding the pool of new recruits increases the chances of finding talented workers. It’s also a matter of survival: Unless unions sign up Brazilians and Dominicans, those workers will flock to non-union jobs and further erode support for the idea of organized labor.

But there is one other reason for the new inclusiveness: a little-known clause in a Boston Housing Authority contract, pushed for by BHA chief of staff Trihn Nguyen, a Vietnamese immigrant.

Expanding the pool of new recruits increases the chances of finding talented workers. It’s also a matter of survival.

It goes like this: The Boston Housing Authority hires a company to do $63 million worth of energy-efficient upgrades. The company agrees to use union labor on the job. It also helps fund a training program to prepare low-income residents for union jobs in construction. And unions who want a piece of this action agree to hire them.

That last crucial component - jobs from the unions - has often been missing from previous efforts to train low-income residents. This time - partly because of the tenacious advocacy of Walsh and Nguyen - unions seem to be on board.

Unions have promised to place all 15 trainees who graduated in November. One - 25-year-old Patrick Pochette, the son of Haitian immigrants - is already working.

The success of the seven-week training program - called Building Pathways - hinges on coordinator Brian Doherty, whose father raised eight kids on carpenters’ union wages. When Doherty wanted to earn money for college, his father helped him join a union too. Now he wants to give others a shot.

“I tell them ‘Somebody made that call for me. I’m going to make that call for you,’ ’’ he said.

There is something beautiful about that. But there is also something worrisome. There will always be more people who need well-paid union jobs than there will be jobs to fill. That phone call is still crucial.

One of the people that Doherty has gone to bat for is Tyiesha Thompson, a 37-year-old African-American mother of three who lives in the Old Colony housing project. Thompson has sold everything from cars to cell phones, but those jobs didn’t give her health insurance or a pension.

A few months ago, she stumbled on a flier advertising Building Pathways. She went through the rigorous selection process and was accepted.

But she almost didn’t take it - she had been offered a catering gig and $10 an hour was hard to pass up. But Doherty convinced her to do the training by telling her she could make $60,000 a year in a union.

For seven weeks, Thompson welded pipes and bent metal and shot rivet guns and drove cranes. Now she is preparing to take the math-heavy electricians’ entrance exam. Alternatively, she hopes Doherty can get her a spot in the next class of plumbers’ union apprenticeships.

“You look at the union, and the color is pretty much white,’’ she said. “But I feel like they are reaching out. . . Maybe they need to fill a quota. But we need these opportunities too.’’

If she gets into the union, she says, she’ll work her way up. She’ll become a leader. Who knows? Maybe one day, she says, she can get one of her sons in.