Recognizing whispered speech and converting it to normal speech creates many possibilities for speech interaction. Because the sound pressure of whispered speech is significantly lower than that of normal speech, it can be used as a semi-silent speech interaction in public places without being audible to others. Converting whispers to normal speech also improves the speech quality for people with speech or hearing impairments. However, conventional speech conversion techniques do not provide sufficient conversion quality or require speaker-dependent datasets consisting of pairs of whispered and normal speech utterances. To address these problems, we propose WESPER, a zero-shot, real-time whisper-to-normal speech conversion mechanism based on self-supervised learning. WESPER consists of a speech-to-unit (STU) encoder, which generates hidden speech units common to both whispered and normal speech, and a unit-to-speech (UTS) decoder, which reconstructs speech from the encoded speech units. Unlike the existing methods, this conversion is user-independent and does not require a paired dataset for whispered and normal speech. The UTS decoder can reconstruct speech in any target speaker's voice from speech units, and it requires only an unlabeled target speaker's speech data. We confirmed that the quality of the speech converted from a whisper was improved while preserving its natural prosody. Additionally, we confirmed the effectiveness of the proposed approach to perform speech reconstruction for people with speech or hearing disabilities. (project page: http://lab.rekimoto.org/projects/wesper )