“It’s kind of Ann Richards-level enthusiasm, in the crowds he gathers,” said Harold Cook, a Democratic strategist in Austin and former executive director of the Texas Democratic Party, referring to the outspoken Democrat who served as Texas’ governor from 1991 to 1995 before losing to George W. Bush. “I haven’t seen a response like Beto O’Rourke is getting in a very long time. Beto has got that thing, that star-power deal that you can’t predict in advance.”

But political analysts say Mr. O’Rourke’s chances are a long-shot at best, despite his fund-raising skills. He remains unknown to many voters, even as he has kept up a hectic travel schedule and visited 228 of the state’s 254 counties. In the Democratic primary last month, Mr. O’Rourke lost a number of counties to two lesser-known Democratic rivals, including the Hispanic-dominated border area of the Rio Grande Valley, parts of East Texas and the northern Panhandle region.

On Monday, Mr. Cruz formally began his re-election campaign at a venue that has become one of his favorite political backdrops — the Redneck Country Club in Stafford, 17 miles southwest of downtown Houston. His speech and his surroundings were plastered with references to his new campaign slogan: “Tough as Texas.” It is meant to evoke the resilience of the state as it reeled from and united in the aftermath of Hurricane Harvey.

At a campaign event on Tuesday night in San Antonio, Mr. Cruz acknowledged Mr. O’Rourke’s success in fund-raising and the challenge for Republicans this November.

“That’s a whole lotta money — there’s no doubt about that — and it has been clear for some time that the hard left is energized and they’re mobilized and they’re angry at the president,” Mr. Cruz said, according to The Texas Tribune. “And we are seeing all across the country, the far left giving millions of dollars to liberal Democrats running for office, and it underscores that Republicans cannot take November for granted.”