Updated at 4:35 p.m.: Revised to include cost estimate for troops on the border

WASHINGTON — Amid mixed reports over whether the Pentagon is about to withdraw troops sent to the U.S.-Mexico border just two weeks ago, Rep. Beto O'Rourke joined forces with Sen. Elizabeth Warren to press Defense Secretary Jim Mattis to end the mission and in the meantime, explain its scope and costs.

Their letter, released Tuesday, questions "the lack of a clear mission for the deployed troops, the cost of this operation, and the appearance that the President is using the military for partisan political purposes."

The letter was dated Monday, the same day the general overseeing the deployment said the troops won't be needed beyond Dec. 15 — a statement that critics took as evidence the mission was a stunt all along.

More than 5,000 service members were deployed to the border for what Pentagon officials have called "an expensive waste of time and resources and a morale killer to boot." Just asked Secretary Mattis why the mission continues, and called for putting an end to it once and for all. — Beto O'Rourke (@BetoORourke) November 20, 2018

O'Rourke and Warren, both of whom may run for president in 2020, sought reassurance from Mattis that troops will not be authorized to shoot unarmed migrants even if rocks are thrown at them, as President Donald Trump had suggested. Rep. Jackie Speier, D-Calif., joined their letter to Mattis.

Trump ordered active-duty troops to the border just 12 days before Election Day and vowed to send as many as 15,000. About 5,900 are deployed. The Associated Press obtained a copy of a report to Congress that said the Pentagon is estimating that the cost of the military's mission on the U.S.-Mexico border will be about $210 million under current plans.

The total includes about $72 million for the 5,900 active-duty troops providing support to Customs and Border Protection, plus $138 million so far for 2,100 National Guard troops who have been performing a separate border mission since April.

The total would grow beyond the estimate of $210 million if the active-duty mission is extended beyond the completion date of Dec. 15.

Most of the troops are in South Texas, far from the migrant caravan in Tijuana, Mexico, south of California.

Trump warned of an impending "invasion" by a "caravan" of impoverished Central Americans making their way on foot thousands of miles through Mexico. Retired military leaders and others saw little justification other than an effort to inflame voters.

The deployment became an instant flashpoint in O'Rourke's failed effort to oust Sen. Ted Cruz.

The El Paso Democrat called it a scare tactic meant to "stir paranoia and fear and anxiety." Cruz welcomed it, saying: "We need to marshal every resource, whether it's Border Patrol, whether it's calling up the National Guard or whether it's calling up the military, but we cannot allow 10,000 people to march across the border."

On Monday, Politico reported that the Pentagon will begin a drawdown as early as this week. Army Lt. Gen. Jeffrey Buchanan, who oversees the mission as head of U.S. Northern Command land forces, said some troops are no longer needed and all should be home by Christmas.

"Our end date right now is 15 December, and I've got no indications from anybody that we'll go beyond that," he told Politico.

The Army insisted Tuesday that no timetable is in place for a full drawdown, however.

Trump touted the deployment and made no indication of his timetable.

"They are proud to be on the border. They are proud to be defending our nation, and we are not letting people in," he told reporters before boarding Marine One and headed to Florida for Thanksgiving. "Our soldiers are doing an incredible job, and if you look at the walls that they're building and if you look at all the barricades that they're putting up, they've done a great job."

Rep. Adam Smith, D-Wash., who will replace Texas Republican Mac Thornberry as chairman of the House Armed Services Committee in January, said the abrupt pullback shows "just how empty, demagogic and racially motivated this political stunt was. It appears to be an admission that there was no justification for the mission in the first place."

Defense Secretary James Mattis defends the deployment of thousands of active duty troops to the border with Mexico Wednesday, calling it "a moral and ethical mission to support our border patrolmen." https://t.co/P3l3YxfD0M pic.twitter.com/nTKh51L2AO — ABC News (@ABC) November 15, 2018

Heading to McAllen last week on an inspection tour, Mattis defended the use of troops.

"It's obviously a moral and ethical mission to support our border patrolmen," he said.

THANK YOU. Our troops deserve to spend Thanksgiving, and the remainder of the holiday season, with their families. https://t.co/9NarWvHaMo — Rep. Vicente Gonzalez (@RepGonzalez) November 19, 2018

Republicans and Democrats have questioned the rationale for the troop deployment, though some Republicans have stood with Trump.

Rep. Michael Burgess, R-Pilot Point, defended the deployment in a Fox News appearance Saturday, saying the troops are needed "because of the onslaught of people that were just being unloaded on this country, and unloaded on our social welfare systems — children that are going to have to be placed in schools. Children whose vaccination status is not known."

He blamed the leaders in Honduras and other Central American countries for allowing the migration.

"We are spending a ton of United States tax dollars on taking care of their children who arrive in this country. I think we should deduct that figure from foreign aid that goes back to those countries," Burgess said.

On CBS' Face the Nation on Sunday, Iowa Sen. Joni Ernst, a Republican and a National Guard veteran, defended the deployments as "an opportunity for real-life training."

"As someone who has commanded troops both in peacetime and in wartime to make sure that they keep their skills sharp," she said. "It allows those Border Patrol agents to actually focus on the law enforcement duties."

The criticism has been intense.

When Trump suggested that he wants troops to shoot at any migrants who throw rocks, as some did when they crossed from Guatemala into Mexico, retired Gen. Martin Dempsey, former chairman of the joint chiefs, chastised him and suggested that such an order would be unlawful.

"A wasteful deployment of over-stretched Soldiers and Marines would be made much worse if they use force disproportional to the threat they face. They won't," he tweeted.

Our men and women in uniform are better trained, better equipped, and better led so they meet any threat with confidence. A wasteful deployment of over-stretched Soldiers and Marines would be made much worse if they use force disproportional to the threat they face. They won’t. — GEN(R) Marty Dempsey (@Martin_Dempsey) November 1, 2018

An op-ed in Monday's New York Times, penned by top national security aides under Bill Clinton and George W. Bush, called the deployment a misuse of the military so egregious that Mattis should have resigned in protest.

"The president used America's military forces not against any real threat but as toy soldiers, with the intent of manipulating a domestic midterm election outcome, an unprecedented use of the military by a sitting president," they wrote.

Mattis noted that President Woodrow Wilson sent troops to repel raids by Pancho Villa and his revolutionaries a century ago, and that the three presidents before Trump each sent the National Guard to the border.

"There's nothing new under the sun," he said last week.

The troops have installed razor wire along the banks of the Rio Grande and at ports of entry, along with other barriers, and provided transportation and temporary housing for Border Patrol personnel. They have not engaged directly in law enforcement.

Sr admin official tells me re troops sent to border: “It’s a paper tiger. A total joke. Of limited operational utility and a waste of our troops’ time. Mattis knows it. Nielsen knows it. Kelly knows it. But that battle was lost with the President. He was hellbent on troops.” — Jake Tapper (@jaketapper) November 16, 2018

On Monday, CNN reported that Trump plans to authorize troops to step in if Customs or Border Patrol face violence, an expansion of the current use of force policies they operate under.

"We have a tremendous military force on the southern border. We have large numbers of people trying to get into our country," Trump said Saturday, adding that he will keep the troops on the border "as long as necessary. They built great fencing. They built a very powerful fence — a different kind of a fence. It's been very powerful. The fence is fully manned. Nobody gets through. And when they're caught, they're not released."