WASHINGTON — The Pentagon is poised to send at least 2,000 more active-duty troops to the southwestern border, Defense Department officials said Tuesday, deployments that have already cost the military hundreds of millions of dollars and thrust the department into the center of the debate over border security and President Trump’s proposed wall.

The acting defense secretary, Patrick Shanahan, told reporters that the United States would be sending “several thousand” additional troops to provide more support for the Department of Homeland Security’s border patrol efforts. Defense Department officials later said that they expect that number to be around 2,000.

That would come on top of the 2,400 troops who are there now, bringing the deployed number at the border close to the high of 5,900 that it reached in the weeks surrounding the midterm elections in November.

The deployment of several thousand military troops to secure the southwestern border will total more than $600 million by the end of the fiscal year in September, Pentagon officials also told the House Armed Services Committee on Tuesday.