Parents and children caught crossing the Mexican border into the United States illegally generally can remain together, the Homeland Security chief said Wednesday, in a partial reversal of previous comments.

Secretary John Kelly also made clear that just about any immigrant in the United State illegally is a priority for immigration enforcement.

"The start point is illegal status and then go to the priorities," Kelly told members of the Senate Homeland Security and Governmental Oversight Committee. "They can go after an individual, under the law, if they are on the list because they are illegal."

Kelly has previously said that young immigrants who took advantage of an Obama administration program that protects them from deportation and allows them to work legally in the United States won't be targeted for deportation unless they let that protection lapse or commit a crime. He did not address that group during his testimony Wednesday.

During the nearly two-hour back-and-forth with senators, Kelly said families caught crossing the border illegally generally would not be separated unless the "situation at the time requires it." He gave as examples the mother being sick or addicted to drugs. But he said separation would not be routine.

Kelly's comment contrasted with earlier pronouncements that his agency was considering separation as a deterrent to would-be border crossers, mostly from Central America.

Last month 1,125 people traveling as families were caught trying to cross the Mexican border into the United States illegally, according to Customs and Border Protection.

Immigrant families and children traveling alone have accounted for hundreds of thousands of arrests at the border in recent years, at times overwhelming federal authorities.

Kelly said he has not issued a written directive outlining the policy to border agents, but has told employees that he must approve any such separations. In a somewhat tense back and forth with Sen. Kamala Harris, D-Calif., Kelly said border agents don't need a written policy because he's given the order verbally.

The retired four-start Marine general said his subordinates know that his orders are to be followed even if they aren't written down. And his leadership style has already improved morale within the agency, he told lawmakers.

"We're changing that; it's already changed," Kelly said, referring to a government survey that found morale at DHS was among the lowest in the government. In a terse exchange with Harris, Kelly suggested his takeover of DHS and President Donald Trump's approach to immigration enforcement almost instantly improved the morale issue within the department.

"The greatest impact in raising the morale in the last 90 days is the work force is now allowed to do their job," the former general said.

Kelly also told lawmakers that a sharp decline in people crossing the southern border illegally was due in part to Trump's immigration policies and widely publicized arrests of immigrants living in the United States illegally. Smugglers have also raised prices to bring would-be immigrants from Central America through Mexico and to the border, he said.

Kelly also took some credit for himself, saying that his direct plea to government officials in Honduras, El Salvador and Guatemala along with conversations with church leaders and others in the region have helped curb the number of people trying to sneak into the United States.

Customs and Border Protection said 12,193 people were caught trying to enter the United States illegally across the Mexican border in March. It was the second straight decline in arrests at the border, a likely signal that fewer people are trying to come into the United States illegally, and the fewest arrests in a month in at least the last 17 years.

Kelly said that decline in the number of people trying to enter the United States illegally won't continue unless his agency gets the resources needed to secure the border.

"It won't last ... unless we do something, again, to secure the border," Kelly said. "The wall. A physical barrier. All we know is that physical barriers do work if they are put in the right places."

Kelly said the wall won't be from "sea to shining sea," but in places where border agents say it would be most effective.

In response to a question from Sen. John McCain, R-Ariz., Kelly said the wall could extend beyond a physical barrier and include a mix of technology including drones.

The prospects for that wall and billions of dollars to pay for it appear dim this year, Sen. Claire McCaskill, D-Mo., made clear Wednesday morning.

"The majority of Senate ... is not going to sign a blank check for a wall we know is never going to be built," McCaskill said.