Travis Kalanick’s decision to step down as chief executive of Uber Technologies Inc. stunned his more than 12,000 employees and rippled through Silicon Valley, but it was the culmination of weeks of maneuvering by some of the firm’s biggest backers to oust the nearly $70 billion company’s co-founder.

Mr. Kalanick was in Chicago on Tuesday to interview a candidate for his top deputy position when he received unexpected news about his future, according to people familiar with the matter.

Matt...