The title link below goes to the complete PDF file of the report, which has a section called Controlling Future Welfare Costs (beginning on print page 26) and contains a couple important points about immigration. (The executive summary may be read here, but it is general and has no discussion of immigration.)
With projected spending in excess of $10 trillion, decision makers should explore ways to slow the growth of future spending. At a minimum, government should take the following nine steps: [...]You can see Rector's earlier Congressional testimony focused on welfare costs: Immigration Hearing 5/10/2007 — Robert Rector (Heritage)
Step #3: Reduce low-skill immigration.
High and disproportionate numbers of both legal and illegal immigrants have low levels of education. For example, one-third of current immigrants lack a high school degree. Around 15 percent ($100 billion per year) of total means-tested welfare spending goes to households headed by immigrants with a high school degree or less. Assuming that this ratio remains similar in the future, lower-skill immigrants will receive around $1.5 trillion in welfare benefits in the next decade. [...]
Step #4: Do not grant amnesty to illegal immigrants.
An estimated 11 million to 12 million illegal immigrants reside in the United States. Except for medical care provided in emergency rooms, these individuals are not eligible for federal welfare benefits. Amnesty proposals, however, would offer permanent residence and citizenship to illegal immigrants, thereby opening the entire welfare system to them. Once an immigrant becomes a citizen, it is unconstitutional to deny him eligibility to receive government benefits that are available to all other citizens.
Given that 50 percent to 60 percent of illegal immigrants lack a high school degree, the long-term cost of providing means-tested welfare to this group would be enormous. Amnesty proposals may delay some of these costs by imposing waiting periods before the amnesty recipient can become eligible for welfare benefits or become a citizen, but welfare use is typically spread over a lifetime, so such delays would not greatly reduce total welfare costs. In addition, upon becoming citizens, these former illegal immigrants would have the right to bring their parents to the U.S. as lawful immigrants. Once here, their parents would likely receive large amounts of welfare aid.
Finally, all amnesty proposals provide illegal immigrants with full access to Social Security and Medicare benefits, which would greatly expand future costs in these two non-welfare programs. Providing amnesty would result in some $2.6 trillion in extra spending in Social Security and Medicare alone. [Obama to Spend $10.3 Trillion on Welfare: Uncovering the Full Cost of Means-Tested Welfare or Aid to the Poor, September 21, 2009]