Conservative, tea-party and libertarian groups have joined liberals in fighting a signature Republican bill in Congress that would crack down on illegal-immigrant workers. The legislation, they argue, would hurt businesses and employees while expanding government regulation.

The bill, by Judiciary Chairman Lamar Smith (R., Texas), would require all employers to use E-Verify, an electronic government database that checks whether new hires are eligible to work in the U.S.

Its prospects had looked good in the GOP-controlled House, where it is to be finished by the Judiciary Committee on Wednesday. Regarded as less controversial than others drafted to tackle illegal immigration, it was seen as able to win bipartisan support in the Senate, too, especially with unemployment high.