Among other things, the bill would eliminate the federal Adequate Yearly Progress (AYP) metric for assessing academic progress. It would also eliminate federally required actions to take against underperforming schools, and eliminate federal teacher qualification standards, in order to give states this authority.

Cantor said the House might also work on the 2014 defense spending bill, which would cut non-war funding by $5 billion compared to 2013 levels.

In his weekly colloquy with Minority Whip Steny Hoyer (D-Md.), Cantor declined to say whether the House would allow an open amendment process for this bill. However, the Rules Committee is expected to approve a structured rule for the bill, in order to avoid amendments on sensitive issues like Syria, Egypt and the National Security Agency.

House aides said GOP leaders believe those issues are best left out of the defense spending bill in order to more quickly get the spending bill approved this year, because some proposals could have the effect of slowing congressional passage of the bill.

On other issues, Cantor said he is hoping the House and Senate can agree to some way forward on the farm bill, which the House passed Thursday over strenuous Democratic opposition because the bill did not include a nutrition title covering the Supplemental Nutrition Assistance Program (SNAP).

"The chairman, the Speaker and other members of leadership are in discussions about how to expedite an agreement on the farm bill," Cantor said. "Certainly, it is our intention ...  to act with dispatch to bring to the floor a bill dealing with the SNAP program. It is our intention to do so."

And on immigration, Cantor told Hoyer that the House has not receive the Senate-passed immigration bill, as that bill contains revenue measures that did not originate in the House. That would create a "blue slip" problem if the Senate tried to push the bill to the House for consideration, although House GOP leaders have already said they will not be taking up the Senate bill anyway.

"It is not correct to say that we have that bill," Cantor said. "There was a tax I believe that was added to the bill, so we do not have that."

— This story was corrected to say the House may take up the education and DOD spending bill.