Rubio's office has also criticized the president's proposals for not including a guest worker program or a plan for dealing with the future flow of immigrants.
Officials say the White House has not set a deadline for when the president would choose to abandon the Senate process and send his bill to Capitol Hill. Senate lawmakers have raised the prospect of drafting a bill next month.
Hanging over the entire immigration debate in Washington is a changed political landscape that gives Hispanics more influence in national politics than ever before. Hispanics made up 10 percent of the electorate in the November presidential election and Obama won more two-thirds of their votes, causing many Republican lawmakers to rethink their opposition to immigration reform.
Immigration advocates have vowed to keep reminding GOP lawmakers of the growing political power of Hispanic voters.
"You can choose not to do it, you can choose inaction, but keep in mind that the Latino community is not going to forget," said Eliseo Medina, an immigration advocate and labor leader at the Service Employees International Union.
"If the Senate blocks it, then get prepared for 2014," he added, referring to next year's congressional elections.
Associated Press writer Erica Werner contributed to this report.
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