"I think you'll see, within the next three weeks, a commitment made first in the context of the Homeland Security appropriations bill," Cornyn said during a briefing.
Cornyn noted that a growing number of non-Mexicans have been apprehended while trying to enter the United States through Mexico, raising concerns that terrorists could be exploiting vulnerabilities in the southern border.
"At a minimum," addressing that problem is "going to take a tremendous increase in the detention facilities that are available," Cornyn said. "We simply have to deal with that. We have no real credible threat of detention and deportation now for the overwhelming majority of people who come to our country illegally."
Cornyn said Senate Homeland Security Appropriations Subcommittee Chairman Judd Gregg, R-N.H., has indicated he is "willing to pull from other sources of homeland security money to deal with this in a credible fashion."
He said Senate GOP leaders and the White House also are committed to improving border security. "We've been talking with the White House, and I know they are very close to arriving at a comprehensive policy to accomplish not just the president's principles for a temporary worker program, but also address the security issues as well," Cornyn said.
He said he expects the White House to provide a "more comprehensive vision of how all this will work" in the next few weeks.
"What the president has heard from a number of members of Congress who are very interested in seeing his program succeed -- people like me -- is that in order for the American people and Congress to accept the idea of a temporary worker program, it has to be coupled with enhanced security," he said.
Cornyn added that he and Sen. Jon Kyl, R-Ariz., plan to introduce a comprehensive immigration package by July 1. Cornyn said he also plans to re-introduce legislation this year to create a "North American Investment Fund" aimed at boosting economic development in Mexico.