During a heated exchange with Nancy Pelosi and Chuck Schumer, Mr Trump said he would be "proud" to shut down the government over border security.
If lawmakers cannot agree on a federal budget, funding for some departments will run out at the end of next week.
Mr Trump has asked to include $5bn (£4bn) in the deal for border security.
But Senator Schumer and Congresswoman Pelosi said they had agreed to extend funding for the Department of Homeland Security at current levels of $1.3bn (£1bn) until 30 September 2019.
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