The 800 National Guard troops sent into Washington last week will soon be augmented by hundreds more, as several states with Republican governors commit to supporting President Trump’s crackdown in the city.

But Army officials appear to be trying to keep the troops on the sidelines of the mission, despite the tough-on-crime image that Mr. Trump has sought to project.

The troops have joined an array of federal agents who appeared on city streets after Mr. Trump declared last week that the federal government was assuming law enforcement responsibility in the capital, which he has falsely claimed is essentially lawless.

The first wave of troops sent to the city all came from the D.C. National Guard, which the president can call out directly. National Guard troops from Louisiana, Mississippi, Ohio, South Carolina and West Virginia will soon also be deployed, according to the governors of those states. National Guard officials said that there were 869 troops in Washington as of Monday night; the Republican-led states so far have pledged 1,000 more.

The Republican governors said they were providing the additional troops at the request of the Trump administration. Gov. Mike DeWine of Ohio said that Army Secretary Dan Driscoll had asked for the extra troops. “When the secretary of the Army asks for backup support to our troops that are already deployed, yes, we will back up our troops,” Mr. DeWine told the Columbus Dispatch.

The number is still expected to grow. But the role of the additional troops appears vague, and the answers to even basic questions, including whether they will be armed, have shifted.

“There is no justification for any deployment of Guard forces in D.C., let alone the deployment of hundreds of Guard forces from multiple states, which smacks of a military occupation of the district,” said Elizabeth Goitein, a senior director at the Brennan Center for Justice at New York University’s law school.

“Local crime is a matter to be handled by local law enforcement,” she added.

The places where the troops have been deployed so far tell part of the story. Most have been seen near the National Mall, large monuments and other tourist-heavy areas.

Army officials said that more would be sent to 10 metro stations, most of which are also near tourist and entertainment sites. They include the Foggy Bottom, Smithsonian, Eastern Market and Waterfront stations.

Near the Washington Monument over the weekend, troops posed for photos with tourists. The National Guard presence, with desert sand-colored vehicles parked near the capital’s most visited tourists spots, is now showing up regularly on social media feeds in posts by visitors to Washington.

The rules of engagement for the troops, at the moment, remain limited to supporting, but not providing, law enforcement. That means that troops are not making arrests, though Army officials acknowledged that could change if Mr. Trump decides that he wants an even more forceful presence.

The scores of officers from federal agencies, including the F.B.I. and the Department of Homeland Security, have linked up with local police on active law enforcement operations.

The question of whether the troops will be armed gets a different answer nearly every day. Last week, Army officials at first said that the troops would not be armed, but would have their weapons “close by.”

Asked to define “close by,” two officials said that weapons could be stashed in vehicles. The officials spoke on the condition of anonymity because they were not authorized to address the issue publicly.

Then, over the weekend, the Guard and the Army issued a new statement that sounded both menacing and vague.

“Guard members may be armed consistent with their mission and training,” Maj. Melissa L. Heintz, an Army National Guard spokeswoman, said in the statement.

But the mission at the moment is to support law enforcement, not to enforce the law, officials said.

As of Monday night, the troops stationed in Washington were still not carrying weapons. Commanders said that in most cases, arms were being stored in the weapons rooms back at the D.C. Armory.

The reason, officials said, was basic security: If armed soldiers frisk someone, for example, that person could lunge for a weapon, escalating the situation.

During deployments in Iraq and Afghanistan, troops carrying out body searches were sometimes unarmed when they were in close quarters with detainees. In those cases, armed troops were usually nearby.

The decision to arm National Guard troops in Washington will be made by operational commanders on a mission-by-mission basis, Army officials said.

“Their presence is focused on supporting civil authorities and ensuring the safety of the community they serve,” Maj. Montrell Russell, a spokesman for the Army, said in the statement.

The National Guard has also started sending military lawyers to work on incoming misdemeanor cases stemming from the deployment of forces, to help relieve the burden on the often understaffed U.S. attorney’s office in Washington, according to two people familiar with the move. At least four of the National Guard lawyers were already working on the cases, with more than a dozen more expected to be reassigned as early as next week, the people said.

While it is not unusual for the executive branch to move lawyers from one agency to another on a temporary basis — detailing, say, lawyers from the Justice Department to serve at the State or Defense Departments — legal experts said it was fairly unusual for military personnel to be placed directly into what amounted to low-level civilian prosecutorial positions. But because the National Guard lawyers serve at the pleasure of their chain of command, they can be sent into the courtroom quickly and easily.

With Republican governors lining up to send in more troops, National Guard officials were scrambling to figure out where the soldiers would sleep once they arrive. Washington has several military bases, but none would have enough room to house hundreds of additional troops.

Most of the deployed D.C. National Guard troops live in the area, so housing for them is not an issue. But troops from Mississippi, West Virginia and other states may end up in the same type of housing that was set up for the 5,000 soldiers who came to town for the June 14 military parade that coincided with the birthdays of Mr. Trump and the U.S. Army. Most of them slept on cots in empty government buildings.

Eric Schmitt and Alan Feuer contributed reporting.