General Electric Co. has cleared the latest hurdle to its new headquarters development by agreeing to alter its ­labor plan for the project after a nonunion contractor complained it was improperly shut out of the high-profile job.

GE last week dropped a requirement that contractors hired to work on the renovation of two older brick buildings in Fort Point abide by collective bargaining rules, a so-called project labor agreement that effectively bars nonunion companies from bidding for the work.

GE’s reversal follows objections to the agreement by a large electrical contractor in Holliston, Wayne J. Griffin Electric Inc.

Griffin had complained last month to MassDevelopment, the public agency that owns the two former Necco Court buildings and is leasing them to GE. A 1999 Supreme Judicial Court decision limits the use of project labor agreements in public buildings to complex construction work, which Griffin said isn’t the case with the GE job.

“It’s an important job for the region, and we want to be involved,’’ said Michael Winters, a lawyer for Griffin, which had positioned itself to sue if the labor agreement wasn’t dropped. “We’re glad that we have this resolved. In a general sense, we’re appreciative of GE and everybody being willing to listen and making the decision for a fair and open competition.’’

The next round of bidding is scheduled for Jan. 12, after being delayed while GE reviewed the legal issues that Griffin had raised. Winters said the work Griffin is targeting is valued at more than $3 million. He said the electrical firm has previously worked with the general contractor for the GE job, Consigli Construction.

GE did not explain why it dropped the labor requirement. Instead, the company issued a statement referring to a partnership it has with an umbrella group of labor unions, the Building and Construction Trades Council of the Metropolitan District.

“The partnership between GE and the Metro BTC reflects a shared commitment to ensuring complex projects like GE Innovation Point are built safely, inclusively, fairly and at a superior level of quality,’’ GE vice president Ann Klee said in the statement.

GE will still use a labor agreement for the anchor of its $200 million headquarters complex, a new 12-story building on adjacent property that the company owns.

GE officials say they plan to move into the renovated brick buildings in mid-2019 from nearby temporary offices, and complete the new structure in 2021.

The two Fort Point buildings are being subsidized under a package of incentives the state and city used to lure General Electric to Boston from Connecticut in 2016. A spokeswoman for Governor Charlie Baker referred questions to MassDevelopment, which is chaired by his economic affairs secretary.

A spokeswoman for that agency said it’s pleased a resolution has been reached that will allow bidding and construction on the redevelopment project to move forward.

Lou Antonellis, business manager for the International Brotherhood of Electrical Workers Local 103, in Dorchester, said he remains confident union electrical workers will be on the Necco job. He said three electrical subcontractors that work with unions are prequalified for the job, although he’s not sure all three will submit bids.

“We feel confident, PLA or no PLA,’’ Antonellis said. “We compete every day of the week for public work, and private work. In the construction industry, if you’re not the low bidder, you might as well be last.’’

Jon Chesto can be reached at jon.chesto@globe.com. Follow him on Twitter @jonchesto.