In a 41-page rulemaking petition to be filed Tuesday with the Department of Homeland Security, the National Day Laborer Organizing Network says that “it is sound policy and consistent with the President’s authority to make a categorical determination to prioritize resources away from the estimated 8 million” people who would qualify for legal status, and possibly citizenship, under a plan approved by the Senate last summer.

AFL-CIO President Richard L. Trumka said in an interview last week that the White House would actually improve its bargaining position with House Republicans if Obama unilaterally suspended deportations. Millions of undocumented immigrants would be allowed to join the public debate, Trumka said, putting more pressure on a party struggling to broaden its appeal with Latinos and Asian Americans.

“If I were president, I would have said the following: ‘It’s a broken system. Except for violent criminals, no more deportations until you help me fix a broken system,’ ” Trumka said.

Legal analysts said the truth probably lies somewhere between the White House’s position and the advocates’ demands. Courts have ruled that the executive branch can employ “prosecutorial discretion” in deciding which cases to enforce in light of limited financial resources.