samaya

“The Lord said: ‘I shall describe to you the characteristics of the ten avatars of Vishnu beginning with the Matsya… The figure of Buddha should be represented as calm, having long ears, a fair complexion, wearing a cloth, seated on a lotus with its petals upwards, and as conferring favor and protection.’” - Agni Mahapurana 49.1–8

Many more references abound in the Shastras. Some examples include:

Harivansha 1.41

Padma Mahapurana 3.252

Vishnu Mahapurana 3.18

Bhagavata Mahapurana 1.3.24, 2.7.37, 11.4.22

Naradiya Mahapurana 2.72

Linga Mahapurana 1.71

Skanda Mahapurana, Vasudeva Mahatmya 18.41

Garuda Mahapurana 1.1, 2.30.37, 3.15.26

Brihat Parashara Hora Shastra 2.1.5–7

Now it is an important point that both Hinduism and Buddhism agree that there have been many Buddhas, not just one. The question has been raised, which Buddha is referenced as an avatar of Vishnu? Is it the most famous Buddha, Shakyamuni Siddhartha Gautama Buddha? Is it Sugata Buddha or Adi Buddha? The answer is, the Shastras clearly reference multiple different Buddhas, referring to different parentages and birthplaces. However, most (not all) of them seem pretty clearly to refer to Siddhartha Gautama, as they reference the Buddha avatar’s father being named Shuddhodana, which was indeed Siddhartha Gautama’s father’s name. For example:

“Agni said: ‘I am describing the manifestation of Vishnu as Buddha, by reading and hearing which one gets wealth… Vishnu, who is of the form of Maya, became the son of Shuddhodana.” - Agni Mahapurana 16.1–2

The Bhavishya Mahapurana is the most explicit: “At this time, mindful of the Kali Yuga, the God Vishnu became born as Gautama, the Shakyamuni, and taught the Bauddha Dharma…”

In addition to these scriptural references, many ancient (and modern) Hindu temples include Buddha images alongside other avatars of Vishnu, with the same reverence.