"If our headquarters staff is cut back to a skeleton, that inhibits our ability to stay on top of a lot of the existing situations and challenges to homeland security right now," he told a news conference, adding this was "not in the public's best interest."
Aides to Senate and House Republicans declined to outline lawmakers' next steps. Some moderate Republicans support removing the immigration provisions or passing a one- or two-month extension of last year's funding levels, which would buy time for a court challenge to Obama's immigration orders.
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Senate Homeland Security and Government Affairs Committee Chairman Ron Johnson on Monday called for an end to the impasse but did not suggest a path forward.
"Responsible members of both parties must work together to find some way to fund DHS without further delay," he said in a statement.
A senior Senate Democratic aide said the party would likely reluctantly accept a short-term funding extension.