WASHINGTON — President Trump’s confrontations with potential Democratic challengers in 2020 have mostly been limited to sniping on Twitter. But on Monday night in El Paso, he will engage in his most direct conflict with a possible rival — former Representative Beto O’Rourke, a native of the city that shares a border with Mexico.

Mr. Trump’s rally in El Paso is his most significant since the midterm elections that delivered the House of Representatives to Democrats. It will provide the president with a backdrop that he will use to again argue for a border wall to stop what he said is a surge of crime and drugs being brought into the country by migrants seeking illegal entry.

“If immigrants are good for the country, the border is secure and walls end lives instead of saving them, then why does the president try to scare us about immigrants and the border and take our land to build a wall we don’t need?” Mr. O’Rourke said on Twitter.

Mr. Trump relishes political combat and this is the first time in the 2020 election cycle that he will be on a dueling, if separate, stage with a Democrat who may try to unseat him. Mr. O’Rourke told Oprah Winfrey that he would make a decision about running for president by the end of February.

It seems a near certainty that Mr. Trump will try to counter the protest in El Paso and its unofficial leader, Mr. O’Rourke. How frontally he criticizes the former congressman, and whether he tries to brand him with a derogatory nickname will be measures of how seriously he takes Mr. O’Rourke’s potential candidacy.

Mr. O’Rourke is the near opposite of the president in tone. During his losing senate campaign last year, he offered himself as a candidate who could bridge divides and wanted to go beyond convenient party labels. That modulated approach is seen as part of his larger appeal. He was able to attract donations nationwide and build a following even as he lost his race.

Now he will face a different test in an opponent whose first instinct is to fight.

Cable news outlets are expected to cover the rallies, and the president almost always takes priority over the words of someone who does not even hold elective office. This time may be different. At a minimum, prepare for a split-screen approach that will offer the freshest soundings of what promises to be a long and bitter presidential campaign.