The Trump administration is expected to announce next week a decision to take off the market all e-cigarettes except those that taste like tobacco and menthol, according to a White House official, revising its previous plan after pushback from conservative groups who argued it could hurt the president’s standing with voters next year.

Aimed at curbing a rise in teenage vaping, the plan has been changed to allow a carve-out for menthol vaping products, according to the official. But it would still be a crippling hit to a market...