WASHINGTON — It’s something most Democrats and Republicans in Congress can agree on — an update to the Bush-era No Child Left Behind education law is much needed and long overdue.

This week, the Senate and House take up rewrites of the 2002 law, with lawmakers seeking to finally resolve a key question they have struggled with for years: how much of a role should the federal government have in ensuring a quality education and boosting achievement for children, poor and affluent alike. Getting enough support to send a bill to President Barack Obama that he’ll sign also may be no easy task.

Even before the floor debate began in the Senate, the White House weighed in late Monday — saying the Obama administration can’t support either the Republican-drafted bill in the House or the bipartisan measure in the Senate. The Senate was scheduled to take up its version Tuesday, the House on Wednesday.

The Senate bill is sponsored by Lamar Alexander of Tennessee, the Republican chairman of the Senate Committee on Health, Education, Labor and Pensions, and the panel’s senior Democrat, Washington’s Patty Murray. It passed the committee unanimously in April. No small feat, agree Alexander, a former U.S. education secretary, and Murray, a former preschool teacher — who had to win over conservatives on the panel like GOP Sen. Rand Paul of Kentucky as well as more liberal members, such as Massachusetts Democrat Sen. Elizabeth Warren.

The measure would keep the reading and math tests mandated in No Child Left Behind — but in a significant move, shift to the states, and away from Washington, decisions about how to use those tests to measure school and teacher performance. It also would expressly prohibit the federal government from requiring or encouraging any specific set of academic standards. That’s a reference to the Common Core standards, which were drafted by the states with the support of the administration but have become a rallying point for those who want to see a reduced federal role in education.

“I have always wanted us to have high national goals and high standards,” Alexander said in an interview. “But in our country, I believe you have to do that state by state, and community by community … you just can’t impose it from Washington D.C. You might get short-term results, but you’re going to get a long-term backlash.”

Murray, ahead of the Senate debate, said the bipartisan bill is an attempt to fix a law that isn’t working. “I’m looking forward to working to improve and strengthen this bill throughout the process and I will be focused on ensuring all kids, especially traditionally underserved students, can learn, grow, and thrive in the classroom,” she said.