Other Democratic amendments are less likely to be accepted. One from Rep. Maxine Waters (D-Calif.) would require HHS to report to Congress after six months describing whether any states are having problems developing the health insurance exchanges without federal help. Rep. Keith Ellison (D-Minn.) will propose a similar amendment today.

Rep. Frank Pallone Jr. (D-N.J.) will propose an amendment requiring a Government Accountability Office study on the benefits of state-run exchanges compared to federally run exchanges. And one from Rep. Peter WelchPeter Francis WelchOvernight Defense: Defense spending bill amendments target hot-button issues | Space Force already facing hurdles | Senators voice 'deep' concerns at using military lawyers on immigration cases Merkley leads Dem lawmakers to border amid migrant policy outcry Lawmakers have sights on middlemen blamed for rising drug costs MORE (D-Vt.) would keep $1.9 billion in funding for states that demonstrate progress toward developing information technology systems used to run state insurance exchanges.

Late Monday, the White House said it would veto H.R. 1213 if it were presented for Obama's signature. However, the bill (like many others House Republicans have passed) is unlikely to advance in the Senate.

The House will also take up H.R. 1214, which would end federal funding for the construction of school-based health facilities. That program runs through fiscal 2013, and would save at least $100 million if repealed.

The White House said only that it would "strongly oppose" this bill, and stopped short of a veto threat.

The House will first need to approve the rule for both bills; the Rules Committee approved a single rule for both on Monday. Final votes on the underlying bills are expected by 6 p.m.

— This story was updated at 9:12 a.m.