Bill Gates schemed to take shares in Microsoft Corp. from his co-founder during the early days of the software company following his partner's treatment for cancer, according to a new memoir by the billionaire co-founder, Paul Allen.

The allegation is part of a critical portrait in the book of Mr. Gates, with whom Mr. Allen formed a friendship in grade school that evolved into one of the iconic partnerships of American business. The book, "Idea Man: A Memoir by the Co-founder of Microsoft," is scheduled to go on...