"We have people for four, five years, and they never leave," Mr. Trump added, with a note of frustration in his voice, as he called on the Congress to act on immigration reform legislation.
"What we have is very simple. We want strong borders and we want no crime," the President said, taking a shot at his critics. "The Democrats want open borders and they don't care about crime."
"The President wants to fix our immigration system completely, not just tinker with it," said White House Press Secretary Sarah Huckabee Sanders.
But it's not clear that Republicans in the House can pass any bill on immigration reform - last week, 41 GOP lawmakers joined with Democrats in voting against a more conservative plan.
Republicans have yet to schedule a vote on what's considered to be a more moderate immigration reform measure, which has drawn only tepid support from the President.
Meanwhile, Democrats continued to press for action on the reunification of families who were separated under a Trump Administration policy started in early May - a policy which the President stopped with an executive order filed last week.
"Trump's now-infamous practice of separating immigrant children from their parents at the border lacks any real plan for family reunification," said Rep. Jamie Raskin (D-MD).
And there were continuing questions today on whether that Trump executive order would spur further legal battles, as a past court settlement does not allow the detention of children for more than 20 days.
"President Trump has created total chaos with his executive order," said Sen. Dianne Feinstein (D-CA).
As the White House brushed aside complaints from Democrats, there was not much way in the strong support being voiced by Mr. Trump for a GOP immigration plan that might be voted on this week in the House.
Asked about the plan at Monday's White House press briefing, Sanders followed the lead of her boss, pointing the finger of blame at Democrats.
"Instead they would rather rant and rave about not letting members of the Trump Administration set foot in public," Sanders said, in part alluding to public heckling of top officials - including herself - as she was asked by a restaurant owner to leave a Virginia establishment on Saturday.
"I was asked to leave because I work for President Trump," Sanders said.
Top Democrats urged their supporters to tone down their attacks on individuals in the Trump Administration.
"No one should call for the harassment of political opponents," Senate Democratic Leader Charles Schumer said on the Senate floor. "That's not right. That's not American."
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