Thousands of union janitors and supporters from nearby states rallied and marched in downtown Boston Saturday — with US Senator Elizabeth Warren as a key speaker — to demand higher pay and more opportunities for full-time work as they negotiate a new contract.

“This city needs you,’’ Warren told the crowd gathered on Boston Common. “These gleaming towers around us, well, they wouldn’t be so gleaming if it weren’t for the people who came in every night, who worked hard, who dusted, who vacuumed, who emptied the trash, who kept it all going for the rest of us.’’

Contracts for about 13,000 janitors who provide cleaning services for places like the Prudential Center, Northeastern University, and the Massachusetts Bay Transportation Authority expire on Sept. 30, said Roxana Rivera, vice president of Local 32BJ of the Service Employees International Union, which represents the workers.

About 70 percent of the workers have part-time jobs, but the union is pushing for more full-time work so more members can qualify for employer-subsidized health insurance, Rivera said.

“It’s still heavily a part-time industry. Employers do that so they don’t have to provide for employer-paid health care,’’ Rivera said in an interview. “We basically are fighting for more opportunities for full-time work so folks don’t have to cobble [together] two or three part-time jobs just to make ends meet.’’

She said workers are also seeking a cost-of-living wage increase. Under the current contract, janitors who work in the Boston area earn $18 per hour, Rivera said.

“The Boston office market is booming,’’ she said. “We believe that needs to translate to working families here in Massachusetts.’’

The employment deal is being negotiated with Maintenance Contractors of New England, which describes itself as an association of the 20 largest building service contractors.

In a statement, spokesman Matt Ellis said the group believes both sides are “engaging in good faith bargaining’’ and are “committed to reaching mutually acceptable terms for a new agreement.’’

“Each side has presented proposals, which are being reviewed and given careful consideration,’’ Ellis said.

Warren told demonstrators their plight was personal to her because her father was a janitor.

“I saw firsthand how hard he worked,’’ she said. “I saw how backbreaking it was and I saw how underappreciated his work was. I also saw how important a paycheck is to keep a family together.’’

Many in the crowd wore purple SEIU T-shirts and carried signs that read “Raise America with Good Jobs’’ in English and Spanish. After the rally, demonstrators marched down Beacon Street into Back Bay, where they proceeded down Newbury Street before turning onto Clarendon Street and then taking Boylston Street back to Boston Common.

Some people traveled from New York, Connecticut, and New Jersey to show their support.

Jose Cortorreal Melendez Jr., a father of six from Providence, said he’s cleaned MBTA subway cars for five years in his job at a private cleaning company.

He said he sometimes cleans up blood, vomit, and hypodermic needles, but doesn’t have the right equipment.

“We want better working conditions,’’ said Melendez, whose comments were translated from Spanish by a union official. “We don’t have the proper materials for the cleaning.’’

He said hours were cut for some of his colleagues and he fears the same might happen to him. Losing full-time work, Melendez said, would jeopardize his health insurance coverage.

Selida Pol, 57, said she cleans facilities at Tufts University School of Dental Medicine through her position with a private cleaning firm. Her contract is being negotiated separately, but Pol said securing a fair contract with health insurance is crucial.

“To have health insurance at my age it’s very important,’’ said Pol, whose remarks were also translated from Spanish by a union official. “It’s as important as having a job itself.’’

Laura Crimaldi can be reached at laura.crimaldi@globe.com. Follow her on Twitter @lauracrimaldi.