The amendment would prevent any of the immigrants legalized under S 744 from achieving permanent legal status until certain requirements are met.
While the Senate debates the immigration bill, labor and community activists in northern California charge that the Immigration and Customs Enforcement agency continues to require local employers to fire hundreds of workers, saying they have no immigration papers.
Boehner's statement is important because it gives his 234-member caucus a virtual veto over what Congress does, regardless of the Senate.
The SAFE Act has harsh features including criminalizing, at the misdemeanor level, the mere act of being in the country without authorization.
On June 13, the House of Representatives voted on a mostly partisan line to prevent the Obama administration from giving undocumented immigrants brought here as children a break.
President Obama insisted the "moment is now" to give 11 million immigrants in the U.S. without documents a chance at citizenship.
Sister Simone Campbell was upbeat on the possibility of meaningful immigration reform.