There are numerous stories regarding the birth of Sage Dattatreya. The story of birth of Dattatreya as the son of Sage Atri is found in the Srimad Bhagavad Purana. Legend has it that Sage Atri performed intense austerities for taking forward his lineage.
Brahma, Vishnu and Shiva appeared before Sage Atri and told him that all of them would be born as his sons.
In due course of time, Anasuya wife of Sage Atri gave birth to Soma who was the essence of Brahma, Dattatreya who was the essence of Vishnu and Sage Durvasa who was the essence of Shiva.
As per Srimad Bhagavad Purana, sage Atri was blessed by Vishnu saying that I have given (data) myself unto you. Thus as the son of Sage Atri, Vishnu was known as Dattatreya.