AUSTIN, Texas—The state will begin efforts to build its own wall along its border with Mexico, Texas Gov. Greg Abbott announced Wednesday, saying the state was prompted to take action since President Biden has pulled back on former President Donald Trump’s border wall plans.

The state will use an initial $250 million of state funds to explore how to build the wall. Mr. Abbott, a Republican, said building a wall along  areas of the border that don’t yet have one would allow state police to arrest and jail immigrants who cross it on trespassing charges. He said he would build more jails along the southern border to hold such people. States don’t have authority to enforce federal immigration laws.

Mr. Abbott’s announcement came days after Mr. Biden’s administration said it would rescind funding for segments of a border wall allocated under Mr. Trump, and would explore returning to landowners property the federal government seized for the project. Mr. Abbott criticized Mr. Biden for not continuing work on the border wall, which had been the central promise of Mr. Trump’s 2016 presidential campaign.

“Because they are not doing it, Texas taxpayers are having to step up,” Mr. Abbott said. Further money, he said, may come from private donations the state is now collecting via a website. Mr. Abbott called a surge of immigration at the southern border an “invasion” that is causing problems for residents there.

The League of United Latin American Citizens, the nation’s largest Latino civil rights organization, indicated in a news release that it intends to fight Mr. Abbott’s move in court. Domingo Garcia, LULAC’s national president, called the effort a waste of taxpayer money and an illegal use of state powers.

“It’s political grandstanding and it’s gutter politics,” Mr. Garcia said.

From the Archives, March 2021

Border cities like Brownsville, Texas, are seeing their resources stretched as they work to manage the growing number of migrant families crossing the U.S.-Mexico border. WSJ’s Michelle Hackman reports. Photo: Verónica G. Cárdenas The Wall Street Journal Interactive Edition

The Texas effort is similar to one launched by Arizona in 2011 to fund border fence construction in that state, launching a website to collect donations in hopes of raising roughly $50 million. The project ultimately raised about $270,000, according to reporting at the time by the Associated Press. The bulk of the money was ultimately awarded to the Cochise County Sheriff’s Office in southern Arizona to help establish a so-called virtual wall consisting of a network of cameras.

A private effort to crowdfund a border wall, the We Build the Wall campaign, raised more than $25 million from hundreds of thousands of donors starting in late 2018. Last year, leaders of the effort, including former senior Trump adviser Steve Bannon, were arrested and charged with defrauding donors, and accused of siphoning the money for their own use. Mr. Trump pardoned Mr. Bannon on his last day in office.

It is a complicated task to build a border wall in Texas. Unlike states such as Arizona and New Mexico, the southern border in Texas is well populated and nearly all property is privately owned. South Texas landowners have generally resisted government seizures of their land, and efforts to do so via eminent domain typically take years. Mr. Abbott said the state would look for private landowners who want to volunteer their property for the project.

The Trump administration built 52 miles of fencing in areas where none had existed before and replaced almost 400 miles of wall along the nearly 2,000-mile U.S.-Mexico border. Little of that was in Texas, where efforts to seize private property slowed the process.

—Alicia A. Caldwell contributed to this article.

Write to Elizabeth Findell at Elizabeth.Findell@wsj.com