On the relationship between the month of Ramadan and Iyyar (alternatively, Torah source for fasting on Ramadan):
شَهْرُ رَمَضَانَ الَّذِي أُنزِلَ فِيهِ الْقُرْآنُ هُدًى لِّلنَّاسِ وَبَيِّنَاتٍ مِّنَ الْهُدَىٰ وَالْفُرْقَانِ فَمَن شَهِدَ مِنكُمُ الشَّهْرَ فَلْيَصُمْهُ وَمَن كَانَ مَرِيضًا أَوْ عَلَىٰ سَفَرٍ فَعِدَّةٌ مِّنْ أَيَّامٍ أُخَرَ يُرِيدُ اللَّهُ بِكُمُ الْيُسْرَ وَلَا يُرِيدُ بِكُمُ الْعُسْرَ وَلِتُكْمِلُوا الْعِدَّةَ وَلِتُكَبِّرُوا اللَّهَ عَلَىٰ مَا هَدَاكُمْ وَلَعَلَّكُمْ تَشْكُرُونَ
“The month of Ramadan is that in which was revealed the Quran; a guidance for mankind, and clear proofs of the guidance, and the criterion (of right and wrong). And whosoever of you is present, let him fast the month, and whosoever of you is sick or on a journey, a number of other days. Allah desires for you ease; He desires not hardship for you; and that you should complete the period, and that you should magnify Allah for having guided you, and that perhaps you may be thankful.” (Surat Al Baqarah 2:185)
Ramadan (رمضان) is the ninth month of the Islamic calendar, and is observed by Muslims worldwide as a month of fasting (Sawm) to commemorate the first revelation of the Quran to the Prophet (pbuh). This annual observance is regarded as one of the Five Pillars of Islam. The month lasts 29–30 days based on the visual sightings of the crescent moon. It is described in biographical accounts and in numerous hadiths
The Jews of Arabia had a similar custom in pre-Islamic Arabia. This custom was part of observing the laws of “Chodosh” (new grains). This law is described in Leviticus 23:14 “And ye shall eat neither bread, nor parched corn, nor fresh ears, until this day itself, until ye have brought the offering of your God; it is a statute for ever throughout your generations in all your dwellings.” This means that no grains – Wheat, Barley, Oat, Spelt, and Rye – could be eaten until the Omer grain offering (Arabic ‘Umra) was brought. This was done on the day after Passover. When the Temple was destroyed, the offering could no longer be brought, so new grains were forbidden to be eaten.
The prohibition of new grains was expanded to all foods including water in the first half of the third century as described in Baba Basra (60b): “Our Rabbis taught: When the Temple was destroyed for the second time (70 CE), large numbers in Israel became ascetics, binding themselves neither to eat meat nor to drink wine. R. Joshua got into conversation with them and said to them: My sons, why do you not eat meat nor drink wine? They replied: Shall we eat flesh which used to be brought as an offering on the altar, now that this altar is in abeyance? Shall we drink wine which used to be poured as a libation on the altar, but now no longer? He said to them: If that is so, we should not eat bread either, because the meal offerings have ceased. They said: [That is so, and] we can manage with fruit. We should not eat fruit either, [he said,] because there is no longer an offering of firstfruits. Then we can manage with other fruits [they said]. But, [he said,] we should not drink water, because there is no longer any ceremony of the pouring of water. (Mishnah Sukhah 4)
The Rabbinic Jews interpreted the words “until this day itself” in the above verse (Leviticus 23:14) to mean that that new grain could be eaten after this DATE had passed even when the Temple was destroyed. The Jews of Arabia interpreted “until this day itself” to mean that grain, as well as all other foods and drink, could be eaten after the DAY had passed, so they fasted during the daytime eating only after nightfall, for the entire month of Iyyar (Ramadan) several centuries before the Hijra.
When the Prophet (pbuh) came, Allah SWT saw fit to reaffirm the Arabian custom (not the Rabbinic version) and make it part of Islam.