Frank Sharry's America's Voice offers "The Anti-Worker Truth About the Anti-Immigrant Lobby" [1]. From their summary:

In recent months, some of the most virulent anti-immigrant Members of Congress have been taking advantage of hard economic times to advance their same, old mass deportation agenda. They argue that blocking comprehensive immigration reform would somehow help the American worker and furthermore, that an unrealistic, multi-billion dollar mass deportation plan would provide instant relief to hardworking Americans in need of good jobs.

But a closer look at the voting records of these Members shows them to be some of the most consistent opponents of legislation to benefit American workers. And analysis of their immigration policy proposals reveals their main goal to be expelling millions of Latinos, Asians, Haitians, Africans, and other immigrants from the United States, not leveling the playing field for all workers and expanding the tax base. When it comes to protecting the American worker, the anti-immigrant lobby simply has no legs to stand on.

1. While they do pretend that comprehensive immigration reform would help Americans, they're also more or less implicitly ceding the point of those Members of Congress, that reducing the number of illegal aliens in the U.S. would help Americans. That's why they're using the deportations false choice: because they can't present an argument that a gradual reduction in the number of illegal aliens would somehow hurt American workers.
2. Few - and I would guess none among the referenced Members of Congress - are calling for mass deportations; more on that in #5.
3. No one with any power wants to expel "immigrants", i.e., those who came here legally; America's Voice is trying to mislead their readers.
4. Part of their report consists of the same old debunked smears of FAIR and related groups, such as by using the "hate group" designations of the Southern Poverty Law Center, not exactly a trustworthy source.
5. Their section on the costs of deportations references the Center for American Progress' "Deporting the Undocumented", a joke study that used a highly-flawed methodology. It also mentions the misleading, business-sponsored Perryman Group study that was briefly mentioned in the second footnote here and the misleading Cato Institute study discussed here. For both of the last, America's Voice tries to pretend they aren't on the same side as cheap labor employers, not revealing the business ties of the Perryman study and of the last saying: Even the conservative Cato Institute has said that "legalization of low-skilled immigrant workers would yield significant income gains for American workers and households."
6. Consider the following:

Sending an out-of-work auto worker and her family in Michigan to pick strawberries in California is not a credible answer to the many Americans desperately in need of good jobs at high wages with good benefits. Rather than promoting a race to the bottom, comprehensive immigration reform would expand labor rights and create a level playing field to ensure better jobs and working conditions for all.

That has an un-American and anti-Mexican subtext, as if only Mexicans and Central Americans are able to pick crops and as if Americans are too good to take bad jobs until the economy improves, even if it involves moving to a different state. The latter is a rather un-American idea.

And, in most cases, Americans wouldn't have to move far at all. Reducing the number of illegal aliens in Michigan would free up jobs for Americans, and likewise with California. And, regarding the fallacy of that "level playing field", see immigration wage floor.

[1] americasvoiceonline.org/pages/
the_anti_immigrant_truth_about_the_anti_immigrant_lobby