Uber Technologies Inc. chief Dara Khosrowshahi is under pressure from the board and investors to find a new finance chief after its leading candidate backed out, adding to the challenges of preparing for an IPO next year.

Mr. Khosrowshahi had hoped to reach an agreement with VMware Inc. CFO Zane Rowe for Uber’s long-vacant role. But Mr. Rowe recently indicated he will turn down the job, according to people familiar with the matter.

Representatives...