Congressional Republicans on Tuesday brushed off President Trump’s threats of a government shutdown over border wall funding, saying that while money for the wall is important, they’re not going to let it send Congress into partisan gridlock.

The annual appropriations process is moving faster than it has in years, and Senate Majority Leader Mitch McConnell said he’s intent on making sure it doesn’t get bogged down in ideological spats. He said he supports Mr. Trump’s goal of a border wall, but his party is not going to slide into a shutdown.

“We’re trying to go through a normal appropriations process that prevents a big event at the end of the fiscal year, which has become all too common around here,” said Senate Majority Leader Mitch McConnell.

Mr. McConnell and House Speaker Paul D. Ryan met with Mr. Trump at the White House last week and the two GOP leaders thought they had convinced the president to cool it on his wall demands. They’re intent on avoiding a shutdown just before voters go to the polls in November.

Mr. Ryan emerged from the meeting to say Mr. Trump was ready to be patient.

But the president has been anything but patient, taking to Twitter on Sunday, Monday and Tuesday to say he’s ready to have a government shutdown unless he sees action on his immigration priorities.

“I don’t care what the political ramifications are, our immigration laws and border security have been a complete and total disaster for decades, and there is no way that the Democrats will allow it to be fixed without a Government Shutdown,” Mr. Trump tweeted.

He continued, “Border Security is National Security, and National Security is the long-term viability of our Country. A Government Shutdown is a very small price to pay for a safe and Prosperous America!”

Mr. Trump is fighting for major policy changes, including eliminating the yearly visa lottery that doles out immigration passes by chance, and curtailing the chain of family migration that allows immigrants to sponsor people outside the nuclear family.

He also wants more than $20 billion to complete his plans for hundreds of miles of new and replacement fencing on the border.

Congress carved out $1.6 billion in the 2018 spending bills. Mr. Trump initially asked for another $1.6 billion for 2019, only to later up his request.

Senators’ current Homeland Security spending bill only includes $1.6 billion, covering the original request. The House version includes $5 billion.

The Homeland Security bill is one of 12 annual spending bills needed to keep the government running, which are due by Sept. 30.

In recent years none of the bills have been done on time and the government has flirted with a shutdown.

But this year many of the bills are moving quickly in both the House and Senate, signaling Congress could at least fund part of the government normally before the deadline — leaving only a few areas in danger of shutting down.

Senate Minority Leader Charles E. Schumer said Republicans must resist Mr. Trump in order to keep on that track.

“They’re going to have to show some strength and tell Donald Trump, ‘if you want to get the government working and not in total gridlock he’s going to have to leave it to the Congress,’” Mr. Schumer said.

The White House said Mr. Trump is committed to getting action on immigration, though spokesman Hogan Gidley took a softer line than Mr. Trump, suggesting a shutdown showdown could be pushed off until after the election.

“He wants Democrats to come to the table and stop playing politics with people’s lives. He said whether a shutdown happens before or after the elections, his focus is getting the problem fixed,” Mr. Gidley said.

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