The Rama Bhujanga Stotra is a hymn to Rama composed by the Advaita philosopher Adi Shankaracharya. Here is what he says in verse 3 of the hymn:
śivo rāma rāmeti rāmeti kāśyām ।
tadekaṃ paraṃ tārakabrahmarūpaṃ
bhaje'haṃ bhaje'haṃ bhaje'haṃ bhaje'ham ॥
I adore shrI rAma, the Unity, the greatest one, whose name is uttered in the ears of the blessed devotees in KAshi at the time of Death by none other than Lord Shiva Himself. This form/name of the Lord (tArakabrahma) is capable of rescuing devotees from the ocean of samsAra.
Now I'm familiar with the notion of Shiva whispering other things, like the Pranava mantra, into the ears of those who die in Kashi. And I'm familiar with the notion of Shiva chanting Rama's name in general. But my question is, what scriptures mention Shiva whispering Rama's name into the ears of those who die in Kashi?
By the way, the words "rāma rāmeti rāmeti" in the second line echo the famous "śri rāma rāmeti rāmeti" verse where Shiva tells Parvati about the glories of chanting Rama's name.
EDIT: I just found a reference to this notion in Tulsidas' Ramcharitamanas:
The might of Rama's name is immeasurable ; so tell the saints, the Puranas and the Vedic commentaries ; the immortal Sambhu, who is the lord Siva, the perfection of wisdom and goodness, is ever repeating it: though all the four groups of animate beings in the world attain to salvation if they die in his city Kasi, yet, O king of saints, it is by the virtue of Rama's name; and therefore Siva in his compassion enjoins its use.
ÉDIT 2: @Vishalprabhulawande points out that in this video, the Shankaracharya of Puri also says that Shiva whispers Rama's name into the ears of those who die in Kashi.