Those drawbacks make it all the more critical for the Republican stronghold of Texas to itself become a selling point in the brothers’ political résumé. To help turn Texas into a swing state, the brothers are supporting President Obama’s former field operatives in a project called Battleground Texas, which seeks to turn out the Hispanic voters who offer Democrats a demographic opportunity. Their hope is that the 2014 Democratic nominee for governor, Wendy Davis, wins a remarkable upset, or at least has a strong enough showing in November to force a Republican presidential candidate to compete in a state that Republicans have long taken for granted.

“That opens up a real opportunity for him,” Ms. Davis said of the mayor after he introduced her last week at a rally in a park framed by freeways. She added that when it came to putting Texas in play in a presidential election, it helped to have “a vibrant leader like Julián being on the ticket.”

The three-term mayor, wearing gleaming black shoes, talked up his signature prekindergarten program in an interview, but grew impatient when questions began to hint at his vice-presidential ambitions. Asked what Democratic Hispanic candidate could counter a Republican Hispanic candidate on the 2016 presidential ticket, he smiled and put sweetener in his iced tea.

“I don’t know,” he said. “That’s a good question.”

The answer?

“Call me in 2016,” he said.

The congressman said his brother or Representative Xavier Becerra of California would be the most obvious Democratic vice-presidential choices if Republicans put a Hispanic on the ticket, but the mayor insisted it was too early to game out scenarios. When it was suggested that one spot seemed taken, he said, “Who’s that, Clinton? Yeah, yeah.” As it happens, the mayor had lunch with former President Bill Clinton in San Antonio in February.

“It was good for Julián because Julián had not really had a chance to be around him much,” said Henry Cisneros, a former San Antonio mayor and Clinton cabinet official who brought Mr. Castro to the lunch and is considered by many to be his mentor. “Even if Julián was not vice president, I think he could very easily be in a Hillary Clinton cabinet.”