Lord Buddha is considered to be an avatar of Vishnu by tradition within Hinduism. While some versions mention Gautam Buddha as being so some other versions opine that Balarama, Krishna’s brother, as being the actual one instead. However there is no clarity on whether or not either is the avatar.
An interesting story, however, exists which supports that Buddha was actually the ninth avatar of Vishnu
According to a legend there are two Buddhas: Sugata Buddha and Gautama Buddha. Vishnu’s avatar was Sugata Buddha. Siddhartha Gautama was born in Lumbini whereas Sugata Buddha was born in Bodh Gaya (Kikata). Gautam Buddha’s mother was “Maya” whereas Sugata Buddha’s mother Name was Anjana.
Lord Vishnu took this avatar to help Lord Shiva kill demon Tripurasura who was creating havoc on earth. Tripurasura had a wife who loved him very much and followed the “Pativrata dharma”. She always prayed for her husband’s good health and long life. Due to her devotion she received a boon that nobody could kill her husband. Unable to do anything Shiva asked Lord Vishnu for help. Lord Vishnu took the avatar of Sugata Buddha which was without any form of clothing to attract her. Once she was going to temple to pray for her husband’s long life. On the way she saw The Handsome Buddha sitting under a tree without clothes. That Buddha was none other than Sugata Buddha. After seeing Vishnu’s handsome body Tripurasura’s wife got attracted to him. She started to slowly forget her husband and her “Pativrata” dharma had been lost . Utilising the opportunity Lord Shiva came and immediately killed Tripurasura and Lord Vishnu once again restored peace on Earth.
Gautama Buddha was a sage, philosopher, teacher and religious leader on whose teachings Buddhism was founded. He is revered by Buddhists as being an enlightened teacher. He shared his insights to help other human beings free from the cycle of rebirth and suffering (moksha). Accounts of his life, discourses and monastic rules are followed by them and have been summarised after his death and memorized by his followers. Various collections of teachings attributed to him were passed down by oral tradition and first committed to writing about 400 years later.
Buddha's portrayal in Hinduism varies in some texts such as the Puranas. He is portrayed as an avatar born to mislead those who deny the Vedic knowledge. In other versions he is seen to have incarnated to end animal slaughter and spread compassion among people.
Buddhists traditionally do not accept the Buddha to be a Vishnu avatar. The adoption may have been a way to assimilate Buddhism into the fold of Hinduism. Much like Hinduism's adoption of the Buddha as an avatar, Buddhist legends too adopted Krishna in their Jataka tales, claiming Krishna (Vishnu avatar) to be a character that Buddha met and taught in his previous births.
Buddhism, “asserts that there is no permanent self or soul in living beings, everything is impermanent” The ignorance or misconception that anything is permanent is considered as a primary source of clinging and suffering and is considered wrong in the understanding of Buddhism.
Buddha endorsed and taught the concept of rebirth which refers to a process whereby living beings go through a succession of lifetimes as one of many possible forms of life, each running from conception to death. In Buddhist thought, however, this rebirth does not involve any soul, unlike Hinduism and Jainism.