WASHINGTON—House Republicans hold a rare weekend meeting today to consider their next steps in a high-drama showdown with the Senate over the terms for avoiding a government shutdown that could begin Tuesday.

A bill to fund federal agencies for several weeks, which Congress must approve by midnight Monday to avoid a partial shutdown, has been approved by the Republican-led House. It was then was passed in a different form on Friday by the Democratic-led Senate.

That leaves House Speaker John Boehner (R., Ohio) and his Republican conference to make the next move in the legislative volleying. In today's session, Republicans will discuss what provisions they want to attach to the spending bill before returning it to the Senate.

Mr. Boehner has said the House won't approve the Senate plan without making changes, a nod to conservatives in his conference who want to use the moment to delay or limit the new federal health law, often called Obamacare.

Federal agencies will have to suspend many nonessential services and furlough workers on Tuesday, the start of the new fiscal year, unless lawmakers can come to an agreement. The odds of a shutdown are rising as House Republicans and Senate Democrats show no sign of breaking their standoff on whether the spending bill can include provisions relating to the health law.