WASHINGTON—Senate Democrats on Tuesday blocked the chamber from considering bipartisan foreign-policy legislation in a bid to pressure Republicans to reopen the government, just hours before President Trump was expected to make a national appeal to end the stalemate on the grounds of national security.

The competing moves, delivered on the 18th day of a partial government shutdown, suggested that neither side is prepared to cede ground over funding for Mr. Trump’s proposed wall along the nation’s southern border.

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