TOKYO—Nintendo Co.’s biggest battle these days isn’t against other game makers. It’s against companies like Apple Inc. that are gobbling up the same parts Nintendo needs to make its hit Switch machine, people in the industry say.

Nintendo 7974, +1.22% has told suppliers and assemblers it hopes to make nearly 20 million units of the Switch device in the year ending March 2018, people involved in the discussions said. Though the company’s official sales target for the year is 10 million, strong demand suggests it can sell many more—if it can make them.