Here is a Gaudiya Vaishnava perspective through scriptural basis. Anything said, in this matter, without the scriptural basis is useless so sorry if it is lengthy but it constitutes only tiny part of what I wanted to say :)
I will begin by quoting some Upanishad verses [All citations from vedabase.com]:
Katha Upanisad 2.2.13 nityo nityanam cetanas cetananam eko bahunam yo
He is the prime eternal among all eternals. He is the supreme living
entity of all living entities, and He alone is maintaining all life.
Svetasvatara Upanisad (3.8–9) says:
vedaham etam purusam mahantam aditya-varnam tamasah parastat tam eva
viditvati mrtyum eti nanyah pantha vidyate ’yanaya
yasmat param naparam asti kiñcid yasman naniyo no jyayo ’sti kiñcit
vrksa iva stabdho divi tisthaty ekas tenedam purnam purusena sarvam
“I know that Supreme Personality of Godhead who is transcendental to
all material conceptions of darkness. Only he who knows Him can
transcend the bonds of birth and death. There is no way for liberation
other than this knowledge of that Supreme Person.
“There is no truth superior to that Supreme Person, because He is the
supermost. He is smaller than the smallest, and He is greater than the
greatest. He is situated as a silent tree, and He illumines the
transcendental sky, and as a tree spreads its roots, He spreads His
Now, who is the Supreme among all living entities? In order to help us understand this Lord Krishna has descended on this planet and performed pastimes that are impossible for any human to perform and boldy declared that He is the Supreme Person. His declarions are intact in Bhagavad Gita. Here are some:
aham sarvasya prabhavo mattah sarvam pravartate iti matva bhajante mam
budha bhava-samanvitah I am the source of all spiritual and material
worlds. Everything emanates from Me. The wise who perfectly know this
engage in My devotional service and worship Me with all their hearts.
mattah parataram nanyat kiñcid asti dhanañ-jaya mayi sarvam idam
protam sutre mani-gana iva
O conqueror of wealth, there is no truth superior to Me. Everything
rests upon Me, as pearls are strung on a thread.
na me viduh sura-ganah prabhavam na maharsayah aham adir hi devanam
maharsinam ca sarvasah
Neither the hosts of demigods nor the great sages know My origin or
opulences, for, in every respect, I am the source of the demigods and
After Krishna spoke 10+ chapters of the Gita, including above three, Arjuna says:
arjuna uvaca param brahma param dhama pavitram paramam bhavan purusam
sasvatam divyam adi-devam ajam vibhum ahus tvam rsayah sarve devarsir
naradas tatha asito devalo vyasah svayam caiva bravisi me
Arjuna said: You are the Supreme Personality of Godhead, the ultimate
abode, the purest, the Absolute Truth. You are the eternal,
transcendental, original person, the unborn, the greatest. All the
great sages such as Narada, Asita, Devala and Vyasa conﬁrm this truth
about You, and now You Yourself are declaring it to me.
However Arjuna wanted a practical proof so that in future fools won't claim that they are Supreme. So Krishna shows His universal form in which He shows everything including all the devatas. While showing the form, Krishna says:
pasyadityan vasun rudran asvinau marutas tatha bahuny adrsta-purvani
O best of the Bharatas, see here the different manifestations of
Adityas, Vasus, Rudras, Asvini-kumaras and all the other demigods.
Behold the many wonderful things which no one has ever seen or heard
Then He also shows the form of four handed Visnu form and finally His beautiful two handed form. Then He says:
bhaktya tv ananyaya sakya aham evam-vidho ’rjuna jñatum drastum ca
tattvena pravestum ca paran-tapa
My dear Arjuna, only by undivided devotional service can I be
understood as I am, standing before you, and can thus be seen
directly. Only in this way can you enter into the mysteries of My
So Lord Krishna, right from the beginning, acted as the Supreme. When He was a couple of days old He killed Putana demon. When He was about 6 years old He lifted Govardhana hill. Inspite of His performing wonderful deeds, specifically saying that He is the Supreme and showing the universal form as well it is not easy to accept this and understand the mystery. That is why He says only by unalloyed devotional service can we understand Him. So for those who have are not surrendered, and not free from the false ego, they cannot engage in His bhakti. And with out being situated in His bhakti it is not possible to accept Krishna as the Supreme.
However as far as the scriptures, as cited above, and Krishna's words are concerned He is the Supreme. And when I say Krishna the Vishnu-tattva is included. But He is the primeval form from Whom other Vishnu-tattvas come as well (Remember Krishna has showed a sample four armed Vishnu form also outside of the Vishva-rupa).