Eight years ago, a group of Irish entrepreneurs led by an ambitious leader, Eoghan McCabe, relocated their software startup, Intercom, from Dublin to San Francisco, aiming to make a splash on tech’s biggest stage. They succeeded, raising more than $240 million in funding from pedigreed venture capital firms and joining an elite class of tech “unicorns”—startups valued by investors at more than $1 billion.

But in recent years, at least a half-dozen Intercom employees have left the company over allegations that Mr. McCabe, its chief executive, made unwanted advances toward female Intercom staffers, according to people who left and others briefed on their departures. The allegations include an incident from 2015 when he was said to have propositioned a young sales staffer after a night of drinking with other Intercom employees and other occasions around that time when Mr. McCabe was said to have slapped the buttocks of a different Intercom employee and touched her thigh at a work party. Multiple people who either witnessed Mr. McCabe’s behavior or heard about it from the women involved told The Information about the incidents.