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But some groups argue, at least publicly, that the House bill won’t necessarily be stronger than the Senate version.

According to some reports, a bipartisan group of seven House lawmakers negotiating a deal may even be looking at a shorter path to citizenship than the one in the Senate agreement.

Rep. Raul R. Labrador, an Idaho Republican who used to be part of that bipartisan group of negotiators, said there is little doubt that the House bill will focus more on security than the Senate version.

“My concern with the Senate bill is that they put the legalization of 11 million people ahead of security. The legalization happens first, and then the security happens second,” he told NBC’s “Meet the Press” program Sunday.

He said he doesn’t want a repeat of what he considers mistakes of implementing the president’s health care law by giving the administration a large amount of power.

Mr. Labrador said it would be a mistake to turn over border security decisions to the Obama administration.

Rep. Michael T. McCaul, Texas Republican and chairman of the House Homeland Security Committee, said on CBS’ “Face the Nation” program that the House also wants to avoid another part of the health care debate, when the House took up and passed the Senate’s bill in its entirety, unchanged.

Some Senate Democrats have predicted that the House will do that with the Senate immigration bill, but Mr. McCaul said that won’t happen.

Earlier this year, he cleared a tough border security bill through his committee on a unanimous bipartisan vote, and he derided the Senate’s plan to add agents and fencing as throwing money at the problem.

“What the Senate just passed was, again, a bunch of, you know, candy thrown down there, a bunch of assets thrown down there to gain votes but without a methodical, smart border approach,” he said. “We want a smart border. We also want a smart immigration plan, something that makes sense.”

Mr. McCaul said his own border security bill could be on the House floor later this month or in September.

Ms. Cadena from the Texas Alliance said she hasn’t given up hope that the House could ease some of the border provisions — particularly if groups pressure representatives at the local level.

“I think that people have pretty much given up that what we got in the Senate is the best we’re going to get, from now on it’s only going to get worse. There might be some truth to that, but those of us who are in the field out in Texas or other parts of the country, it’s too soon to make that determination,” she said.

“At the end of the day, all politics is local. We’re trying to influence some of these members who may be in swing districts or may be in significant Latino population.”