The Trump administration has given 36 countries, most of them in Africa, a Wednesday deadline to commit to improve vetting or face a ban on their citizens visiting the United States.A weekend diplomatic cable sent by the State Department instructs embassies and consulates in the 36 countries to gauge their host countries’ willingness by Wednesday to improve their citizens’ travel documentation and take steps to address the status of their nationals who are in the United States illegally.

Trump’s administration is considering an expansion of its imposed travel ban — potentially prohibiting citizens from 36 more countries from entering the United States, according to a State Department...