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The White House argues that it has taken sufficient steps on immigration enforcement and says it is time to turn attention to legalization.

But Republicans in border states have challenged Mr. Obama’s claims of security.

Texas Attorney General Greg Abbott told The Washington Times on Tuesday that City Hall in El Paso, Texas, was struck by gunfire from across the border in Juarez, Mexico, last year. A college in Brownsville, at the other end of Texas’ 1,200-mile border with Mexico, “had bullets fly across its campus from cartel members across the border,” he said.

“It is factually incorrect for anyone to suggest that the border is secure and is not being penetrated, and that has to be the initial linchpin of any comprehensive immigration reform that is going to take place in the country. Without that, there really is no immigration reform,” Mr. Abbott told The Times.

Republicans working on an immigration bill in the Senate have insisted that any pathway to citizenship for illegal immigrants be created only after the border is deemed secure and after the country has established a better system to weed out illegal workers.

The president has said he wants illegal immigrants to have a definite path to citizenship that isn’t tied to any “triggers” such as border security.

Mr. Obama met Tuesday with two of the Republicans involved in that bipartisan group, and Sens. Lindsey Graham of South Carolina and John McCain of Arizona emerged to say they were encouraged by the talks.

The president has said the senators must finish writing their bill quickly or else he would send his own version to Capitol Hill and demand a vote on it.

Tom Howell Jr. contributed to this report.