The Judiciary Committee, trying to get its work on the bipartisan measure completed by the end of the week, also on Tuesday fought off proposals that threatened the fragile compromise reached by the so-called Gang of Eight, the bipartisan group of senators who drafted the legislation, the Los Angeles Times reported.
The committee approved an amendment offered by Sen. Charles Grassley, R-Iowa, that would require customs officials to be notified immediately when a student visa is terminated. A student charged with obstruction of justice in the Boston Marathon case had been allowed to enter the United States in January even though his student visa was no longer valid.
"This will plug a loophole in the tragic Boston Marathon bombings," Sen. Charles Schumer, D-N.Y., said.
Senators voted down amendments from Sen. Jeff Sessions, R-Ala., that would have required a biometric exit visa system, saying costs and logistics would be prohibitive.
The Times said much of Tuesday's markup concerned levels of legal immigration who would be allowed to enter the United States, particularly for high-skilled workers.
Sessions sought to limit the number, citing depressed pay and loss of jobs for U.S. workers while Sen. Ted Cruz, R-Texas, sought to bump up the number of visas allowed for workers in high-tech fields by 500 percent. The committee rejected both amendments.
A group of conservative House Republicans protested outside the Capitol, with some citing findings of the conservative Heritage Foundation that said immigrants would be a net cost to taxpayers, the Times said.
"America cannot afford to open massive immigration floodgates any more than it can support an amnesty plan," Rep. Mo Brooks, R-Ala., said.