Republicans generally argue that PLAs essentially require contractors to use unionized labor, which raises the cost of construction projects. As a result, they oppose a 2009 Executive Order from President Obama that they say is seen as a requirement to use PLAs. Democrats, in contrast, say the order only suggests that agencies consider PLAs as part of construction projects.

But the issue is splitting Republicans, just as it did last year. Rep. Michael Grimm (R-N.Y.) on Thursday evening proposed an amendment to strike language in the bill that prevents PLAs from being required.

Members debated Grimm's amendment for more than an hour. Several supporters of it said they oppose the underlying bill because it is seen by some as prohibiting the use of PLAs, while opponents said they believe Obama's order requires the use of PLAs.

Rep. Jeff FlakeJeffrey (Jeff) Lane FlakeMcSally takes hard line on immigration in Arizona primary Flake threatens to limit Trump court nominees: report Poll: McSally holds 14-point lead in Arizona GOP Senate primary MORE (R-Ariz.) proposed language in the bill that prevents PLAs from being required, and said that language is meant to keep the issue neutral, and allow some PLAs when they make sense, depending on the details of the project. "It's neutrality," Flake said. "That's what this bill returns to."

In the end, Grimm offered to work on new language that both sides can support, and suggested that some agreement could be coming shortly.

"You feel that the language of the President is somehow restricting non-union shops from bidding," Grimm said. "And I… strongly feel that the language in your amendment absolutely prohibits the use of PLAs.

"I think what we're both looking for is neutrality, but the language on either side is not working, so we need to come up with a way to make this neutral."

In the meantime, however, a vote on Grimm's amendment as it stands now could also occur late Thursday evening.

Last year, the House had a similar fight over language in the Veterans Affairs bill. That bill had language prohibiting the use of funds to enforce the Obama administration order. The House ultimately voted 204-203 in favor of an amendment eliminating that language, and revealed the split within the GOP on this issue — 27 Republicans voted for the amendment.