WASHINGTON (CNN) -- House Minority Leader John Boehner said Tuesday he has "significant concerns" about parts of Senate immigration bill, the Ohio Republican's first comments on the bipartisan measure and the latest sign it faces a questionable future.

"America is a nation of immigrants, but we are also a nation of laws -- those laws must be obeyed and enforced," Boehner said in a statement. "The Senate agreement appears to recognize that additional border security measures and more effective immigration law enforcement must come before any other issues are addressed, but I have significant concerns about parts of the Senate proposal -- particularly provisions that would reward illegal immigrants who have consistently broken our laws."

Boehner's comments come the same day the Senate is set to begin a general discussion about the measure and debate an amendment sponsored by Sen. Byron Dorgan, D-North Dakota, that aims to eliminate the bill's call for a temporary worker program -- a policy labor unions claim will depress American workers' wages.

Several more amendments from both sides of the aisle are expected to be introduced throughout the week ahead of the Senate's Memorial Day recess.