According to the opinion closest to the consensus of the scholars Rasulullaah Sallallaahu alayhi wasallam passed away in the month of Rabiul awwal. The day of demise was definitely Monday by consensus.
However the scholars are divided regarding the actual date of demise. There are three main opinions :
1) 12th Rabiul awwal - This has been preferred by majority of the scholars like Ibn e Ishaaq, Ibn e Sa'ad, Ibn e Hibbaan, Ibn e Abdul Barr, Ibnus Salaah, Nawwi, Zahabi and Ibn ul Jawzi. (Footnotes of Lami' 485 - 486/8)
2) Musa Ibn e Uqba, Laith, Khuwarzimi, Ibn e Zuber - 1st Rabiul awwal
3) Abu Mikhnaf, Kalbi and preferred by Allamah Suheili - 2nd Rabiul awwal. Hafiz Ibn e Hajar has also declared Abu Mikhnaf s view to be most reliable. (Fathul Baari 735/7)
Allaama Suheili Rahmatullahialayh and his followers have declared the 12th to be problematic. It's a proven and agreed upon fact that the 1st of Dhulhjjah 10AH was on a Thursday and the demise of the Prophet Sallallahu alayhi wasallam was definitely on a Monday. Now even if we regard Dhulhijjah, Muharram and Safar, all to have been of 30 days or all to have been 29 days or some to have been 29 and some to be 30 days, 12th Rabiul awwal does not fall on a Monday in any case.
Replying to this Mazeril and Ibne Kathir Rahmatullahialayh say that it's possible that all three months had been 30 days and sighting of the moon might have been on different days in Makkah and Madeenah. Going by this possibility the 1st of Dhulhijjah in Makkah was on Thursday while in Madeenah it fell on Friday. Consequently the wuqoof of Arafah etc was according to the date of Makkah. After returning to Madeenah the dates used were according to the moon sighting of Madeenah. In this way 12th rabiul awwal does fall on a Monday. (Albidayah wan Nihayah)
This is the reasoning of Mazeril and Ibn e Kathir Rahmatullahialayhima; but Hafiz Ibn e Hajar Rahmatullahialayh disagrees with it as it would mean assuming four months to have been 30 days consecutively.
Lending weight to the opinion of 2nd Hafiz Ibn e Hajar says that it's possible that the original wordings were thani shahri Rabi il awwal(2nd). Due to errors this then became thani ashar (12th). This became so common that people started using it without any contemplation. According to Allama Suheili, in this case Dhulhijjah, Muharram and Safar, all will have to be regarded to be 29 days, then only 2nd Rabiul awwal will fall on Monday.
As for the opinion regarding the 1st Rabiul awwal it has been supported by Ibn e Jarir's narration reported by Ibn e kathir that between the revelation of the verse sura'e Maidah (9th Dhulhijjah) and the demise of Rasulullaah Sallallaahu alayhi wasallam there were 81 days. (Tafseer Ibn e Kathir 13/2) In this case 1st of Rabiul awwal will fall on a Monday if, out of Dhulhijjah, Muharram and Safar, Two months are taken to be of 29 days and one month of 30 days - and this is very common occurrence.
To sum up, out of the three main opinions regarding the date of Rasulullaah Sallallaahu alayhi wasallam demise the 2nd seems to be most probable as adopted by Allama Suheili and his followers and authenticated by Ibne e Hajar Rahmatullahialayhim.
ALLAH KNOWS BEST
Mufti Umar Faruq Desai Luharvi
(Shaykhulhadith Darul Uloom London)