This claim came from a narration in some Sunni books from ‘Urwah Ibn al-Zubair from his aunt Aisha that the Holy Prophet (SAWA) passed away while he was putting his head on the chest of Aisha. This narration is a fabricated narration and it contradicts with authentic narrations in Sunni and Shia books . Ibn Abbas, the well known companion, denied and refuted this narration.
as it is in Tahtheeb al-Tabaqaat al-Kubra (page 402) : Abu Ghatfan asked Ibn-Abbas, “did you see the Prophet when he passed away putting his head on the chest of anyone?”
Ibn Abbas replied: “the Prophet passed away when his head was on the
chest of Ali ibn Abi Talib”. The narrator asked him, that: “‘Urwah told
me that Aisha said that the prophet passed away when his head was on
her chest.” Ibn Abbas said: “do you have sense? Can you believe it? By
Allah, the messenger of Allah, passed away and his head
was on the chest of Ali Ibn Abi Talib.”
In the well known Sunni book Al Tabaqat al Kubra (vol 2 pg 260) Ibn Sa’ad said: The prophet passed away while his head was on the chest of Ali-ibn-abi-Talib.
Ibn Asaakir, the well known Sunni scholar, narrated in his book
Tareekh Dimashq (vol 3 pg 15 narration number 1028) that the mother of
Jumai’ ibn ‘Omair and his aunt both went to Aisha and asked her about
Ali. Aisha said: “you are asking me about whom? About a man who put
his hand, when the prophet was dying, and the soul of the prophet came
out on his hand and he wiped his face with it.”
This narration has also been mentioned in Majma’ al Zawa’id by al Haythami (vol 9 pg 112)
and also mentioned by al Zamakhshari in Rabi’ al Abraar.
The false narration claiming hat he Prophet put his head on the chest of Aisha was narrated by a well known enemy of Imam Ali (AS), ‘Orwah ibn al Zubair, who fabricated this false allegation because of his enmity against Ali Ibn Abi Talib (AS) . 'Orwah Ibn al Zubair fought against Imam Ali (AS) in the Bttle of Jamal in Basra and tried with his brother Abdullah to kill mam Ali (AS).
How can Muslims believe an enemy of Ali and leave what Ali (AS) himself stated? Ibn Abi Al-Hadeed narrated that Ali (AS) himself said(ولقد قبض رسول الله وان رأسه على صدري ) “verily the Messenger of Allah (SAWA) passed away when his head was on my chest.”
Nahjul Balagah, (sermon 197).
In Musnad Ahmad bin Hambal (vol 6, pg 300) Umm Salama,
one of the wives of the Holy Prophet (SAWA), she said: “I swear by
whom I swear, that Ali was the nearest to the Holy Prophet (SAWA) and
the last person who was with the Holy Prophet (SAWA) when the Holy
Prophet (SAWA) passed away, he was alone with Ali giving him secrets
and talking to him alone.”
This is also mentioned in al Musannaf (vol 7 pg 494) also in Kanz al
Ummal (vol 13 pg 146 Hadeeth number 36459) and also in Tareekh Madinat
Dimashq and also in al Bidaya Wannihaya by ibn Katheer (vol 7 pg 397)