WASHINGTON—President Trump renewed a threat to shut down the government unless Congress provides money for a wall at the southern border and enacts new curbs on immigration, reviving a battle he said last month should wait until Republicans gained more solid majorities after the November elections.

“I would be willing to ’shut down’ government if the Democrats do not give us the votes for Border Security, which includes the Wall!” Mr. Trump wrote on Twitter on Sunday. “Must get rid of Lottery, Catch & Release etc. and finally...