House Speaker John Boehner is aiming to have various chamber committees finish work on immigration legislation by the Fourth of July, and would like a House vote by August, reported Politico.com.

The speaker wants to see movement on either several small immigration bills or a more comprehensive one before the Senate votes on its measure, Politico reported. Senate Majority Leader Harry Reid, a Nevada Democrat, scheduled a cloture vote on the immigration bill for Tuesday.

The Senate is expected to vote on the full bill, called the Border Security, Economic Opportunity and Immigration Modernization Act, by July 4.

A bipartisan group in the House has been working on a bill, but has encountered stumbling blocks on achieving a consensus on parts of the measure, leading one of its members –Rep. Raul Labrador, a Republican from Idaho– to drop out. Labrador said he left the so-called “Group of Eight” over the issue of access to health care programs for undocumented immigrants who pursue a legal status under a reformed system.

“The [GOP] leadership’s plan is to allow the bipartisan group to release its legislation and closely monitor how it is received by House Republicans,” the site reported.

Boehner, from Ohio, will be watching to see if House Republicans – the GOP has a majority in the chamber – who include a conservative group, complain that the bipartisan group’s plan goes too easy on undocumented immigrants.

The Senate bill tightens border security and provides a pathway to legal status for undocumented immigrants who meet a strict set of criteria. It also expands guest worker visas and makes E-Verify –a federal program to ensure that prospective hires are eligible to work in the United States– mandatory for employers.

Some Republicans in the House, as well as some in the Senate, such as Ted Cruz, have said they will not agree to the provision regarding a pathway to legal status, calling it amnesty. But defenders of the provision say it is not amnesty because it does not simply forgive wrongdoing — it requires that those who came or stayed here illegally pay fines and wait more than a decade to get permanent legal status.

Also, over the weekend New Hampshire Republican Kelly Ayotte said on “Face the Nation,” a CBS show, that she is supporting the Senate’s bipartisan bill.

Ayotte drew particular attention to the controversial pathway to legal status and said it is “tough but fair.”

Sen. Rand Paul, a Republican from Kentucky, is scheduled to be the keynote speaker Wednesday at a forum on immigration hosted by Latino center-right organizations that have expressed support for comprehensive reform.

But some of the most vociferous critics of reform proposals are not backing down.

In an ABC News interview, Ted Cruz said President Barack Obama is the main obstacle to immigration reform.

"A path to citizenship is the most divisive aspect of this bill and the White House is insisting on it,” he said, adding that the measure “is designed for it to sail through the Senate and then crash in the House to let the president go and campaign in 2014 on this issue.”

The Heritage Foundation’s political arm, Heritage Action, is pushing for a rejection of the Senate measure.

On the other end, Senate Judiciary Committee Chairman Patrick Leahy said Monday that the Senate should work through differences to reach a consensus on reform.

“The legislation before us is the result of a bipartisan group of Senators who came together and made an agreement,” Leahy said of the Senate bill. “If Senators who have come together to help develop this bill keep their commitments, I have no doubt that we will be able to end this filibuster and pass this fair but tough legislation on comprehensive immigration reform.”

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