Senate Majority Leader Harry ReidHarry ReidBill O'Reilly: Politics helped kill Kate Steinle, Zarate just pulled the trigger Tax reform is nightmare Déjà vu for Puerto Rico Ex-Obama and Reid staffers: McConnell would pretend to be busy to avoid meeting with Obama MORE (D-Nev.) set up a procedural vote on a House bill that would allow companies not to treat veterans as full-time employees under the ObamaCare employer mandate.

On Thursday, Reid filed cloture on the motion to proceed to H.R. 3474, the Hire More Heroes Act of 2014, meaning the first procedural vote on that measure could take place next Tuesday or Wednesday.

Under ObamaCare, employers with more than 50 full-time employees must provide healthcare, but this bill would allow employers to hire veterans and not count them toward that threshold.

The bill says veterans could be exempted from the full-time employee count if the federal veterans health program — TRICARE — covers them. Supporters argue this will help businesses cope with the healthcare mandate and encourage the hiring of veterans.

The House passed the bill on a 406-1 in March.

Sen. Roy Blunt (R-Mo.) introduced a Senate companion measure last month, but Reid is using the bill as a vehicle for the Senate tax credit extender bill.

Reid has resisted holding votes on tweaks to the healthcare law, but vulnerable Democrats up for reelection have insisted it will help their chances in November.

Republicans are likely to insist on amendment votes to the bill, putting final passage in the Senate at risk.

— This article was updated at 3:30 p.m. on May 9, 2014.