WASHINGTON—House Speaker Nancy Pelosi faces a narrowing set of options to avert growing calls within her Democratic caucus for the initiation of impeachment proceedings against President Trump.

Mrs. Pelosi has fended off such calls for months, counseling patience and assuring lawmakers that the full suite of oversight tools in the House’s arsenal would be the best course to hold Mr. Trump to account for efforts to curtail special counsel Robert Mueller’s investigation of Russian interference in the 2016 election. She has also...