Reliably red states such as Arizona and Georgia already could be slipping out of Donald Trump’s hands, but the Hillary Clinton campaign now believes it can make a real play for another grand prize: Texas.

Speaking to volunteers in Austin, Sen. Tim Kaine said the Clinton campaign believes it can win Texas and is “very serious” about mounting a full-scale effort to move the state into the Democratic column for the first time since the 1976 presidential race.

“This team, the Clinton-Kaine team, is very serious, we’re very serious about Texas because we know the kind of work you can do,” the Virginia senator and Democratic vice-presidential nominee told campaign workers. “We’re going to go after Texas. It’s big, its complicated and it’s hard, but we’re serious about it. The way we will win is because of what you are doing right now.”

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Former President Jimmy Carter carried Texas in 1976, but the state has gone Republican in every presidential election since.

But with this year’s race already defying many political norms, even Texas seemingly could be in play.

There’s been scant polling of the state so far, but the surveys that have been done show Mr. Trump’s lead to be anything but insurmountable.

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A University of Texas poll released in late June gave Mr. Trump an 8-point advantage, though that survey came out long before the party conventions. Mr. Trump’s support nationally and in battleground states has dwindled over the past few weeks, and Republican-leaning states such as Utah, Georgia and Arizona now are toss-ups or, in some cases, are actually leaning toward Mrs. Clinton.

Texas still is considered to be leaning Republican, but the state’s large Hispanic population and some GOP and independent opposition to Mr. Trump could make the race tighter than originally expected.