The agency representing some of the region’s largest building maintenance contractors said it is confident it will reach a new contract with some 13,000 janitors before the end of the week, avoiding a walkout that could threaten thousands of office buildings in the Boston area.

Meanwhile, the Service Employees International Union, Local 32BJ,which represents the janitors, is amping up threats of a strike at a number of high-profile rallies, including one scheduled for noon Thursday in Boston with several religious leaders.

Maintenance Contractors of New England, the consortium representing 20 of the largest building service contractors in the area, called the strike authorization “unnecessary’’ and reiterated in a statement that significant strides have been made at the bargaining table.

“I don’t believe the strike vote has any real bearing on the work being done at the bargaining table. It seems to be a perfunctory step the union takes to energize its members,’’ said Matt Ellis, a spokesman for the contractors.

He declined to say whether there was a contingency plan in the event of a strike.

With a couple of days to go, the union said there were still significant issues the two sides did not agree on, including a proposal by contractors to exclude from the contract janitors who work in buildings of 100,000 square feet or less, which would affect about 25 percent of the membership.

Daily negotiations have been ongoing since Sept. 22, but the janitors voted Saturday to authorize a strike if an agreement isn’t reached by the end of the week.

Ellis said the contractors have put an offer on the table for an annual increase of 2.24 percent over four years for janitors in Boston’s central business district, which would bring wages to $19.45 per hour by 2020.

He added that a proposal to provide comprehensive family health coverage at no cost for all full-time employees has been accepted, pending final approval. Under the current contract, full-time employees pay $280 a month for family health insurance, with the remainder paid by the contractor.

Roxana Rivera, vice president of Local 32BJ, said the union wants the contractors to extend the offer of higher wages, longer shifts for part-time workers, and converting more jobs into full-time positions for janitors in buildings in Cambridge and additional ones in Boston.

About 70 percent of the janitors work part time, many just three or 3½ hours per night, Rivera said. She wants the contractors to agree to a minimum of four-hour shifts.

“We’re in the last couple of days here. A lot can happen as the deadline comes near, [and a strike] is more of a reality,’’ Rivera said. “If we don’t move jobs to full time now, we don’t know when we’re going to do that because the economy is really strong.’’

This is the fourth consecutive contract negotiation since 2002during which the union janitors, a mostly immigrant and part-time workforce, have approved strike actions. Although they threatened strikes each time, they walked off the job only in 2002, for 30 days.

Katheleen Conti can be reached at kconti@globe.com. Follow her on Twitter @GlobeKConti.