In a particularly frustrating development for Republican leaders, they are racing to protect two House seats they had already spent tens of millions of dollars defending in special elections: one anchored in suburban Atlanta’s Cobb County and the other in central Ohio. In the Georgia seat, Republicans say a combination of robust Democratic turnout — propelled by Stacey Abrams’s campaign for governor — and a heavy financial investment by Mr. Bloomberg has put them unexpectedly on defense.

Tilting the political map further, Mr. Bloomberg’s group is also financing ads in two other conservative-leaning suburban districts long viewed as relatively safe by Republicans: one held by Representative Jaime Herrera Beutler of Washington, the other a seat in the suburbs of Jacksonville, Fla., that former Representative Ron DeSantis vacated to run for governor.

Republicans are straining to keep up. Just in the last few days, Republican committees have spent millions on Ms. Herrera Beutler and Mr. Mast of Florida — both of whom hold center-right, outer-suburban districts — as well as for a couple of conservative open seats in Virginia and Florida. At the same time, they have added additional funding into a Virginia Beach-based district and seats in southern Minnesota, in suburban Des Moines and around Topeka, Kan.

“We’re now seeing opportunities that we didn’t think would be there,” said Representative Ruben Gallego, an Arizona Democrat, adding: “I do feel confident that we’re going to get the majority and then some.”

Former Representative Lynn Westmoreland of Georgia, a Republican who served as vice chairman of the party’s campaign committee, said Republicans faced races that were “tightening up” across the board, especially in diverse suburbs and districts full of right-of-center “soccer moms.” He said Republicans stood a chance to hold the House if some of those moderate women returned to the fold.

The difference between a narrowly held Democratic House, or even one with the barest of Republican majorities, will most likely turn on about 20 districts that Mr. Trump won comfortably but not overwhelmingly and that contain a mix of new developments as well as smaller, older communities. Many of these seats were gerrymandered by Republican state legislators to protect incumbents, but that was in the pre-Trump era, when Republicans could more reliably count on the votes of moderate women.

“They have an assumption about that voter and their fidelity to the Republican Party that is not true under Donald Trump,” said Rahm Emanuel, the Chicago mayor and former House Democratic campaign chief, of voters he called “anti-Hillary but never pro-Trump.”