Houthi forces battled their way southward toward the strategic city of Aden on Tuesday, heightening the prospect of outside military intervention by Saudi Arabia to protect the country’s embattled president.

President Abed Rabbo Mansour Hadi is holed up in Aden, where he has been attempting to rally domestic and foreign support for his efforts to retake control of the capital San’a from the Houthis. He has urged Saudi Arabia, the regional powerhouse, to come to his aid against the Iranian-backed Houthis, Saudi media reported...