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James Talarico, the Democratic nominee, has been amassing campaign cash as Senator John Cornyn and Attorney General Ken Paxton still battle each other.

Senator John Cornyn of Texas holds a significant fund-raising edge over his hard-line Republican challenger, the state attorney general Ken Paxton, with four weeks to go before early voting begins in their runoff.

But both fell short of the campaign cash amassed this year by James Talarico, the Democrat in the Senate race, according to a New York Times analysis of Federal Election Commission data filed by the campaigns on Wednesday.

Since the primary election last month, Mr. Cornyn raised more than $6.5 million across three funds controlled by the campaign and an independent political action committee.

Mr. Paxton brought in more than $1.9 million in that same time frame, with nearly half raised by a political action committee backing his candidacy.

During that period, Mr. Talarico appears to have done far better.

His campaign and a pro-Talarico political action committee raised more than $5.8 million from donors giving at least $200, according to the analysis of post-primary donations. Including smaller contributions, Mr. Talarico’s campaign said, his total fund-raising since the primary reached more than $10 million. (The two Republican candidates received just over $400,000 in smaller donations since mid-February.)

Mr. Cornyn and the groups supporting him led in total cash left to spend as of the end of March, reporting a total of more than $10 million, nearly three times as much as Mr. Paxton and about $1 million more than Mr. Talarico.

While Mr. Talarico has been able to focus his campaign fund-raising on the general election, Mr. Cornyn and Mr. Paxton remain locked in a primary runoff battle until May 26.

The Republican candidates were pushed into a runoff after neither was able to win a majority of the vote in the March primary. The race has become deeply personal: Mr. Paxton has accused Mr. Cornyn of being insufficiently conservative, and the senator has highlighted the allegations of corruption and adultery against the attorney general.

Both have courted the president’s endorsement, which he has so far withheld.

Mr. Paxton said in a statement that the numbers showed his campaign was entering the May runoff “in an even stronger financial position than we did in March.”

The winner of the runoff will face Mr. Talarico in November. Mr. Talarico, a state representative who has catapulted to national fame, hopes to become the first Democrat elected statewide in Texas in more than 30 years.

Mr. Talarico’s campaign pointed to his fund-raising success in a news release, calling it the most ever raised in a U.S. Senate race in the first three months of a year. Since January, his campaign has brought in more than $27 million. Add in the political action committee supporting Mr. Talarico and that total jumps to $35 million — about $13 million more than Mr. Cornyn and Mr. Paxton raised across all of their funds and committees over the same period.

Mr. Cornyn’s campaign suggested Mr. Talarico’s massive fund-raising haul had come from out-of-state donors and said the incumbent senator was the better bet to beat the Democrat in November.

“Democrats nominated their strongest candidate for U.S. Senate, and Republicans must nominate John Cornyn, their strongest candidate for U.S. Senate to ensure Texas stays red,” said Matt Mackowiak, a spokesman for Mr. Cornyn’s campaign.

Mr. Talarico’s campaign said he received contributions from 246 of the state’s 254 counties and that most were $100 or less. Mr. Talarico’s campaign has been focused on unity, and his brand of compassionate Christianity has targeted voters from all parties.

“We’re bringing Democrats, Republicans, and independents together to end billionaire control over our politics and bring down costs for families across our state,” Mr. Talarico said in the news release.

Top donors to Mr. Talarico’s political action committee included some out-of-state benefactors such as Reid Hoffman, the Washington venture capitalist and LinkedIn co-founder, who donated $1.5 million. Stephen Mandel, a hedge fund manager from Connecticut, and Dr. Lisa Primus of New York gave $500,000 each.

Don Henley, the guitarist for the Eagles, was also on the list.

Mr. Paxton’s contributors included Wallace Hall Jr., a former regent of the University of Texas System who fought removal from the board after accusing its flagship university of wrongdoing in admissions.

Among the donors to the pro-Paxton group was Terry Lacore, a North Texas businessman who has attracted attention recently in Texas political circles because of a stay in a Washington hotel that was booked by Mr. Paxton’s state agency during President Trump’s second inauguration. Mr. Lacore donated $100,000.

The state comptroller has been auditing the episode, which led to the departure of two senior agency employees this year. Mr. Paxton’s office issued a report, saying the matter was the result of “mistakes and errors” in following agency procedures that “created an unacceptable appearance of impropriety.”

Mr. Cornyn’s campaign contributors included Harlan Crow, the longtime Texas Republican donor; the banker Charles R. Schwab; Karl Rove, the Republican political strategist; and George W. Bush, who listed his occupation on the federal form as “former president.”