U.S. government officials are considering stepping up sanctions against Venezuela by targeting its vital oil industry, although an embargo against Venezuelan crude oil imports into the U.S. is off the table for now, people familiar with the deliberations say.

The measures could be announced as early as Monday, the people say, after a vote Sunday pushed by embattled Venezuelan President Nicolás Maduro to elect a special assembly that will rewrite the constitution. Venezuela’s opposition is boycotting the vote, fearing the assembly...