It must by definition fall within either the time period leading up to Passover (commonly called Lent), or from Passover to Pentecost or from Pentecost to Day of Atonement.
Wade Cox of the Christian Churches of God believes it is the time period from Passover leading up to Pentecost. A Jewish Rabbi also believes it is during the period, Sefirat haOmer. This is calculated based on - 10 Muharram, a fast day, in the Islamic calendar, thought to be The Day of Atonement.
Ramadan has during the last 10 days, the Night of Power where the fate of people is determined. In the Hebrew festival, The Day of Atonement is the night where the fate of people are determined for another year.
About Day of Atonement: Yom Kippur—the Day of Atonement—is considered the most important holiday in the Jewish faith. Falling in the month of Tishrei (September or October in the Gregorian calendar), it marks the culmination of the 10 Days of Awe, a period of introspection and repentance that follows Rosh Hashanah, the Jewish New Year. According to tradition, it is on Yom Kippur that God decides each person’s fate, so Jews are encouraged to make amends and ask forgiveness for sins committed during the past year. The holiday is observed with a 25-hour fast and a special religious service. Yom Kippur and Rosh Hashanah are known as Judaism’s “High Holy Days.”
About Night of Power: Most of the Muslims believe that Laylat al-Qadr is to be the night when the Quran was first revealed to Muhammad from Allah. As per the Quran (Sura Qadr) Angels come down on the earth with all things (destiny of the people for the coming year). "The angels and Jibreel descend in it by the permission of their Lord for every affair" (97:4) Similarly Lailatul Qadr' is to be found in the last ten odd nights of Ramadan but mostly on the 19th or 21st or 23rd or 27th night of Ramadan.
Is the Night of Power the Day of Atonement? We cannot be 100 per cent certain but they have some strange similarities.
Islam believes the Night of Power is one of the last 10 days of Ramadan. The Day of Atonement is the 10 day of Tishri, ending the 10 days of Awe.
One Jewish Scholar believes though Lag BeOmer, the day the Zohar is to be the basis for Ramadan.
From the article The Islamic Jewish Calendar:
And the fast of Ramadan to be based on the Jewish Sefirat haOmer which among other things is a time of mourning for the hundreds of thousands killed after Bar Kochba’s failed revolt in 135 CE.The Jews also keep a fast on 10 Tevet which falls in December. This would place Ramadan near to the Day of Atonement. It is interesting the Prophet Muhammed does not know about the fast on 10 Muharram, if it being 10 Tevet, would be an additional Jewish fast. The Prophet would know about the Day of Atonement and when it fell. Since the Day of Atonement is a commanded fast, the most holiest Day to the Jews and Israel. Jewish aristocracy ruled the Middle East in those days with a great number of Jews in the area the Prophet must have been familiar with at least some of the Jewish days.
The Shorter Encyclopaedia of Islam says: "The starting-point for a comparison between the Islamic and the Jewish Calendar is the optional fast-day Ashura, the tenth day of Muharram, which is an Islamic continuation of the Day of Atonement”.6 G. Widengren says: “If Muharram is comparable with Tishri, then the month of Ramadhan is comparable to the Jewish month of Sivan, the month of the Jewish Feast of Weeks. Islamic tradition lays down that that it was on the nights of Ramadhan, … Lailat Al-Kadr … that Muhammed received the revelation of the Koran. … there is a clear parallel between the circumstances in which Moses received the Torah and those in which Muhammed received the Koran”.
From the same article:
There are traditions found in Sunni books to the effect that the Prophet on migrating to Medina found the Jews fasting on the 10th of Muharram.30 He asked them why, and was told: "It is an auspicious day; it is the day when God delivered the children of Israel from their enemy; and, therefore, Moses fasted on that day [because of the broken tablets]." The Prophet said, "I am worthier of Moses [more ‘Jewish’] than you are." Thereupon, he fasted on that day and ordered [the Muslims] to fast. The importance of this fast to the Prophet may be judged from another tradition narrated in al-Sahih of al-Bukhari: "The Prophet ordered a man from the [tribe of] Aslam: Announce to the people that whoever has eaten should fast the rest of the day, and whoever has not eaten should fast [the whole day], because today is the 'Ashura [10th day of Muharram]." In another hadith ‘Ubayd b. 'Umayr said: "Al-Muharram is the month of God. It is the beginning of the year. It is used as the beginning of the era. In al-Muharram, the Ka'bah is clothed, and money is coined. There is one day in al-Muharram on which repenting sinners are forgiven." This “day of forgiveness” on the 10th of the first month of the year was apparently Yom Kippur.
Further study needs to be undertaken on the exact location of Ramadan in the Hebrew calendar, but two possibilities seem plausible, either leading up to Pentecost from Passover or from Pentecost to the Day of Atonement. The Prophet Muhammed is unlikely to be ignorant of the Day of Atonement considering his knowledge of the Sabbath.