A key component of the Apple Watch made by one of two suppliers was found to be defective, prompting Apple Inc. to limit the availability of the highly anticipated new product, according to people familiar with the matter.

The part involved is the so-called taptic engine, designed by Apple to produce the sensation of being tapped on the wrist. After mass production began in February, reliability testing revealed that some taptic engines supplied by AAC Technologies Holdings Inc., of Shenzhen, China, started to break down over...