EL PASO, Texas—Beto O’Rourke has run for three different offices by campaigning against incumbents he painted as out of touch with constituents, a message that will have to change if he gets into the crowded 2020 Democratic presidential primary.

In his narrow loss to Texas GOP Sen. Ted Cruz last year, Mr. O’Rourke built a potent, national political brand by holding town halls in each of the state’s 254 counties and live-streaming much of his activity on the internet. It was a similar playbook to the one he used when running for the El Paso City Council in 2005 and for Congress in 2012.

“His fundamental strength is authenticity,” said David Axelrod, the strategist for former President Obama’s campaigns.

But if Mr. O’Rourke jumps into a presidential primary that could include more than 20 other Democrats, saying he is better than the incumbent isn’t likely to be a strong enough argument to win the nomination and the chance to take on President Trump in the general election, his friends and political allies say.

“If he ran for president, it would be much different,” said Steve Ortega, a longtime O’Rourke friend who in 2005 was elected with him to the El Paso City Council. “He’s not going to take anyone by surprise. He’s no longer the insurgent. He’s one of the front-runners. It’s a different deal.”