Is Sai Baba an incarnation? If so, of Shiva, Vishnu or Rama?

Question

Some people say Sai Baba is an avatar of Shiva while others say it's either Vishnu or Rama.

What is the official position (if any) on this?

Does Sai Satcharitra mention anything about this?

Answer 1

What is the official position (if any) on this?

Official position can be subjective as Scriptures don't mention him as He lived in early 20th century. But i'll mention opinion of Sai Satcharitra on this topic.

Sai Satcharitra from official Shri Saibaba Sansthan Trust, Shirdi mentions Sai Baba as Avatara of Dattatreya at multiple places.

Chapter 1 of Sai Satcharitra says:

(8) And lastly, to his Guru Shri Sainath – an Incarnation of Shri Dattatreya, Who is his sole refuge and Who will make him realize, that Brahma is the reality and the world an illusion; and incidentally, to all beings, in whom the Lord God dwells.

Chapter 36 says:

Shirdi folks must be infinitely superior and more fortunate than we; for, Baba has played, laughed, talked and lived with you for so many years. I think that, your store of good merits must be infinite. Sai is our Datta (Lord Dattatreya). He gave me a seat in the steamer and brought me here, and thus gave proof of His omniscience and omnipotence.”

Chapter 43 says:

One day, during Baba’s absence a boy who was sweeping the floor, took it up in his previous Incarnation of god Dattatreya, Shri Narsimha Saraswati of Gangapur. His passing away is only an outward aspect, but in reality He pervades all animate and inanimate things and is their Inner Controller and Ruler.

So, Sai Satcharitra reiterates at multiple places that Sai is Avatara of Sri Dattaterya.

Answer 2

I am also answering this question based on Shri Sai Satcharita for brevity.

Yes, Sai Baba is an incarnation of Dattatreya.

Since it is already mentioned in this answer that Sai Baba is an avatar of Dattatreya. I am providing some more information regarding the same. There is a parampara (lineage) of Dattatreya and the Sai Baba is the fifth avatar of Dattatreya in that lineage. The sequence of avatars is as follows

1. ShriPaadha ShriVallabha
2. Shri Narasimha Saraswati
3. Sri Manikya Prabhu Maharaj
4. Swami Samarth Maharaj of Akkalkot
5. Sri Sai Baba of Shirdi

Although I am currently not providing all the statements that confirm the complete lineage as it makes the answer extraneous, I will provide the statements from the Shri Sai Satcharita that confirm that Sai Baba continues the Datta parampara after Swami Akkalkot Samarth Maharaj.

A devotee of Akkalkot Maharaj, by name Bhai Krishnaji Alibagkar, worshipped the photo of Akkalkot Maharaj. He once thought of going to Akkalkot (Solapur district), take the darshan of the Padukas (footprints) of the Maharaj and offer his sincere prayer there, but before he could go there, he got a vision in his dream. Akkalkot Maharaj appeared in the vision and said to him, “Now, Shirdi is my resting place, go there and offer your worship!” So, Bhai changed his plan and came to Shirdi, worshipped Baba, stayed there for six months and was happy. As a reminiscence of this vision etc., ......

[The Story of Padukas (foot-prints) under the Neem Tree, Chapter 5]

Later, in the same chapter, Baba also says that he is Akkalkot Maharaj himself

Bhai Krishnaji was originally a devotee of Akkalkot Maharaj. He had come to Shirdi at the installation of the Padukas, in Shaka 1834 on his way to Akkalkot. He wanted to go to Akkalkot, after taking the darshan of Baba. He asked Baba’s permission for this. Baba said, “Oh, what is there in Akkalkot? Why do you go there? The Maharaj of that place is here, Myself.” Hearing this, Bhai did not go to Akkalkot. He came to Shirdi off and on, after the installation of the Padukas.

and you can read chapter 26 for the similarity between both of them. You can also read from here and here for more details.