WASHINGTON—A partisan clash over President Donald Trump’s border wall is heightening chances of a government shutdown next weekend as members of the Trump administration and congressional Democrats drew sharp lines in the sand on Sunday.

Homeland Security Secretary John Kelly, speaking on CBS’s “Face the Nation,” said it is likely that Mr. Trump will insist on including money to fund a border wall between the U.S. and Mexico in a bill this week to fund the federal government for the rest of the fiscal year. “He will do the...