The White House is pushing for Congress to take up an aid package for state and local governments, marking the administration’s first effort to break up the president’s jobs proposal into individual provisions for votes.

The Senate turned back a modified version of the president’s jobs proposal on a 50-49 procedural vote last week. In a statement after the vote, President Barack Obama said the administration would work with Senate Majority Leader Harry Reid (D-Nev.) to move the legislation in pieces.

Speaking with reporters this afternoon, White House spokesman Josh Earnest and economic policy aide Jason Furman indicated that Obama first wants a vote on approximately $35 billion for aid to help keep teachers, firefighters and police officers at work.

The president hinted he would prioritize the aid in his weekly address Saturday.

“I’m urging Members of Congress to vote on putting hundreds of thousands of teachers back in the classroom, cops back on the streets and firefighters back on the job,” he said.

Earnest said that the administration would call on lawmakers to “pass the bill this week,” even though the House is on a weeklong recess.

Senate Democrats had not fully determined the way forward Sunday afternoon, but Reid told reporters on Wednesday that the individual components of the package could be offset through tax increases on people earning more than $1 million per year. Senate Democratic leaders previously proposed a 5.4 percent tax increase for upper-income tax filers to pay for the $447 billion cost of the full proposal.

Democrats could seek to set up a test vote on the standalone aid package by proposing it as an amendment to a “minibus” appropriations measure that the chamber is expected to consider Monday.