Under the legislation outline offered by a bipartisan group of senators last week, that pathway would only exist if the border is deemed to be appropriately secure. But lawmakers did not outline the specific metrics they would use to gauge border security, nor did they explain who would be responsible for judging whether those goals had been accomplished. Under one proposal, a group of elected officials in border states would be given that responsibility.

The White House initially balked at such a requirement, saying it feared that coupling the pathway to citizenship to border security requirements would leave undocumented immigrants in limbo. Democrats, led by Sen. Charles SchumerCharles (Chuck) Ellis SchumerSchumer: GOP efforts to identify FBI informant 'close to crossing a legal line' Patients deserve the 'right to try' How the embassy move widens the partisan divide over Israel MORE (D-N.Y.), have proposed that the Department of Homeland Security instead be granted the power to decide whether goals had been met.

“What we’ve proposed is that the DHS secretary, whomever it is, will have final say on [whether] whatever metrics we propose are met,” Schumer said. “We think those metrics will be quite objective.”

Some Republicans are likely to balk at granting Napolitano such power. But with her trip this week, the Homeland Security secretary is looking to project a tough stance on border security, with plans to highlight the increased number of agents patrolling the border and new technologies being deployed by the federal government.

The president will also look to continue the push on immigration, with a meeting Tuesday at the White House with labor and business leaders intended to promote his immigration reform effort.