The Heritage Foundation, a conservative think tank, is set to release a study this week on the costs of immigration.
"The study you'll see from Heritage this week presents the staggering costs of another amnesty in our country and the detrimental effects long-term that that will have," DeMint said.
DeMint's comments came as defenders of the bipartisan reform measure, including Sen. Marco Rubio (R-Fla.) have for weeks been trying to win support for the bill from conservatives.
Sen. Patrick Leahy (D-Vt.), the chairman of the Judiciary Committee, which has begun hearing on the bill, was hopeful Sunday that it could win passage.
"I think we can get it passed," he said on NBC News's "Meet The Press."