WASHINGTON — House Republicans made progress on Tuesday steering some of their most conservative members away from a course that could shut down the government next week, a prospect that would badly damage the party as it prepares to take control of Congress in January.

Though the outcome remained unpredictable, Speaker John A. Boehner and his deputies were already counting votes for a two-part plan that they presented in a closed-door meeting in the Capitol.

The first step would be to allow a largely symbolic vote on legislation to dismantle President Obama’s executive action last month that delayed the deportation of millions of illegal immigrants. The second would be to fund the government through the current fiscal year, which ends Sept. 30, 2015, except for agencies that spend money to enforce Mr. Obama’s immigration action, like the Department of Homeland Security.

Congress would then revisit funding for those agencies early next year when Republicans are in control of both the House and the Senate and in a stronger position.