Some consider Krishna to be an avatar of Lord Vishnu but others worship him as the supreme godhead.
I read once in Vishnu Purana that Krishna is 'a very tiny part of Lord Vishnu.'
Does anyone know where this is said in the Vishnu Purana or other Hindu scripture?
I think you are referring to the first few verses of Viṣṇu Purāṇa, Book 5, Ch. 1 where both sages Maitreya and Parāśara refer to Kṛṣṇa as aṃśāṃśasaṃbhūti (incarnation of a part of a portion) of Viṣṇu:
maitreya uvāca
nṛpāṇāṃ kathitaḥsarvo bhavatā vaṃśavistaraḥ |
vaṃśānucaritaṃ caiva yathāvadanuvarṇitam |aṃśāvatāro brahmarṣe yo 'yaṃ yadukulodbhavaḥ |
viṣṇostaṃ vistareṇāhaṃ śrotumicchāmi tattvataḥ || ViP_5,1.2 ||cakāra yāni karmāṇi bhagavān puruṣottamaḥ |
aṃśāṃśenāvatīryorvyāṃ tatra tāni mune vada || ViP_5,1.3 ||
Maitreya: You have related to me a full account of all the different dynasties of kings, and of their successive transactions. I wish now to hear a more particular description, holy Rishi, of the portion of Vishńu that came down upon earth, and was born in the family of Yadu. Tell me also what actions he performed in his descent, as a part of a part of the supreme, upon the earth.
śrīpāraśara uvāca
maitreya śrūyatāmetadyatpṛṣṭo 'hamiha tvayā |
viṣṇoraṃśāṃśasaṃbhūticaritaṃ jagato hitam || ViP_5,1.4 ||
Parāśara: I will relate to you, Maitreya, the account which you have requested; the birth of a part of a part of Vishńu, and the benefits which his actions conferred upon the world.
This is a matter of interpretation:
Since, we have direct statement in Bhagavata which separates Krishna from amsas (According to Amara-kośa, a noun followed by tu has no relation to anything that precedes it.(tv-antāthādi na pūrva-bhāk).)
SB 1.3.28:
ete cāṁśa-kalāḥ puṁsaḥ kṛṣṇas tu bhagavān svayam
All these incarnations are either plenary portions or parts of the plenary portions of the puruṣa-avatāra. But Kṛṣṇa is the Bhagavan Himself.
(or) All these are either portions or minute portions of the Puruṣa, but Kṛṣṇa is Svayam Bhagavān. (Bhagavata Purāṇa 1.3.28)
The Vishnu Purana verses(5.1.2-4):
aṁśāvatāro brahmarṣe yo’yaṁ yadu-kulodbhavaḥ | viṣṇos taṁ vistareṇāhaṁ śrotum icchāmy aśeṣataḥ|| cakāra yāni karmāṇi bhagavān puruṣottamaḥ | aṁśāṁśenāvatīryorvyāṁ tatra tāni mune vada || maitreya śrūyatām etad yat pṛṣṭo ’ham idaṁ tvayā | viṣṇor aṁśāṁśa-sambhūti-caritaṁ jagato hitam ||
which may be generally translated as:
“[Maitreya said to Parāśara:] “O Brahmaṛṣi, I want to hear extensively and completely about the well-known Avatāra of Viṣṇu’s aṁśa who appeared in Yadu’s dynasty. O sage, tell me about the deeds He, the Lord, the topmost Person, performed after He descended on Earth in that line with an aṁśa of an aṁśa.” [Parāśara said:] “Maitreya, you’re asking me about the deeds, auspicious for the world, of He who appeared with an aṁśa of an aṁśa of Viṣṇu. Listen.””
or
MAITREYA.— I wish now to hear a more particular description, holy Ṛṣi, of the portion of Viṣṇu that came down upon earth, and was born in the family of Yadu. Tell me also what actions he performed in his descent, as a part of a part of the supreme, upon the earth.
Parāśara said:— I will relate to you, Maitreya, the account which you have requested; the birth of a part of a part of Viṣṇu, and the benefits which his actions conferred upon the world.
Later in the same chapter (5.1.59-65), Lord Vishnu says:
When Brahmā had ended, the supreme lord plucked off two hairs, one white and one black, and said to the gods, “These my hairs shall descend upon earth, and shall relieve her of the burden of her distress.
Let all the gods also, in their own portions, go down to earth, and wage war with the haughty Asuras, who are there incorporate, and who shall every one of them be destroyed. Doubt not of this: they shall perish before the withering glance of mine eyes. This my (black) hair shall be impersonated in the eighth conception of the wife of Vasudeva, Devakī, who is like a goddess; and shall slay Kansa, who is the demon Kālanemi.” Thus having spoken, Hari disappeared; and the gods bowing to him, though invisible, returned to the summit of mount Meru, from whence they descended upon earth.
Further we see in the same chapter (5.1.76), Lord Vishnu says
I will myself become incarnate in the eighth conception of Devakī; and you shall immediately take a similar character as the embryo offspring of Yaśodā.
In Summary we have four statements:
1) Bhagavata verse says Krishna is Svayam Bhagavan.
2) First statement of Vishnu Purana says 'Krishna is tiny part of Vishnu'
3) Second statement of Vishnu Purana of same chapter says 'Krishna is Vishnu's black hair'
4) Third statement of Vishnu Purana of same chapter says 'Krishna is Vishnu himself'
Now, one has to harmonize all these statements.
In the light of Bhagavata verse(kṛṣṇas tu bhagavān svayam), the same set of verses are explained by Srinatha Chakravarty in his commentary to Bhagavata as follows:
The explanation of the above controversial words is this, sequentially:
(1) aṁśāvatāraḥ (the Avatāra of an aṁśa) signifies: aṁśānāṁ nārāyaṇādīnām avatāraḥ yasmāt saḥ, “He from whom there is an Avatāra of aṁśas such as Nārāyaṇa,”
(2) aṁśāṁśena avatīrya (after descending with an aṁśa of an aṁśa) means: aṁśaḥ brahmādayaḥ teṣām aṁśena yādava-rūpeṇa saha avatīrya, “after He descended with the Yādavas, who are aṁśas of aṁśas such as Brahmā,” since it will be said: bhavadbhir aṁśair yaduṣūpajanyatām, “[Brahmā told the gods:] You should take birth as aṁśas among the Yadus” (Bhāgavatam 10.1.22), and
(3) viṣṇor aṁśāṁśa-sambhūti-caritam (the deeds of He who appeared with an aṁśa of an aṁśa of Viṣṇu) is understood as follows. At first, viṣṇoḥ caritam means śrī-kṛṣṇasya caritam (Śrī Kṛṣṇa’s deeds). What were they like? They were aṁśāṁśa-sambhūti, which means: aṁśānāṁ brahmādīnām aṁśāḥ yādavāḥ teṣāṁ samyak samīcīnā bhūtiḥ sampattiḥ yasmāt yatra vā, “either
(A) because of which there is the complete prosperity of the Yādavas, who are aṁśas of aṁśas such as Brahmā; or (B) in which the prosperity of the Yādavas, who are aṁśas of aṁśas such as Brahmā, becomes complete.”
