The bipartisan Elementary and Secondary Education Act overhaul will likely hit the floor of the U.S. Senate for debate late this month or early July, said Sen. Lamar Alexander, R-Tenn., chairman of the education committee and one of the bill’s co-authors.

While the ultimate decision will be made by Majority Leader Mitch McConnell, R-Ky., Alexander said Tuesday morning at an event hosted by National Journal that the federal K-12 reauthorization could get called to the floor as early as the week of June 22.

“I think we’ll be successful. Maybe if we’re able to pass it with a big vote, they’ll be able to pass it too,” Alexander said, referencing the U.S. House of Representatives, which would also need to approve the bill before it could become law. The House was widely expected to pass a separate bill in February, but the effort faltered, thanks in part to conservative opposition.

Importantly, the measure would also need the president’s signature. Alexander said that he and his staff have been working in close concert with the president and his staff on substance of the reauthorization, adding that there are policies in the bill that the administration both likes and would like see changed.

The administration has recently expressed concerns about accountability, specifically ensuring safeguards for subgroups of students, including low-income students, students with disabilities, English learners, and minority students. In particular, U.S Secretary of Education Arne Duncan, has said the Senate bill should beef up language dealing with how states monitor their poorest-performing schools.