According to the Midrash, Abraham underwent ten trials, one of which was the circumcision, and the most severe was the order to sacrifice his son Isaac – something which greatly troubled the Danish philosopher Søren Kierkegaard, and caused him to write his classic Fear and Trembling. These trials are listed in the Mishna, Abot 5.3, and in parallel passages elswhere (Abot de Rabbi Natan,33.2, Midrash Psalms, 18.25, Pirqe de Rabbi Eliezer 26.)
Another trial was his being cast into a fiery furnace by the wicked Nimrod, just as were thrown much later the three "Holy Children" by King Nebuchanezzar in the book of Daniel. This trial is detailed in the Babylonian Talmud, Pesahim 118a:
…when the wicked Nimrod cast our father Abraham into the fiery furnace, [the angel] Gabriel said to the Holy One, blessed be He: 'Sovereign of the Universe! Let me go down, cool [it] and deliver that righteous man from the fiery furnace.' Said the Holy One, blessed be He, to him: 'I am unique in my world, and he is unique in his world: it is fitting for Him who is unique to deliver him who is unique.'
But as a reward to the angel for his good intentions, he is promised that he would deliver Daniel and his friends at a later time.
The Nimrod story is found also in the Babylonian Talmud, `Erubin 53a: "Nimrod …ordered our father Abraham to be cast into a burning furnace."
There is a legend that Abraham's father was a maker of idols. Abraham decided that this was foolish, broke the idols, and placed a stick in the hand of the biggest idol, explaining to his father that the big idol had broken the others. When his father said this could not be, he asked why one would venerate such an object. These materials are expounded in the Arabic text, Abraham filling the role of a religious debater. The "ten days" also are taken to refer to Abraham's trials.
In Midrash Bemidbar Rabba 2.12 there is a parable of a man who traveled ten days in the wilderness without finding any kind of shelter. He then arrived at an oasis. He ate the fruit of the tree, drank from the spring, and enjoyed the shade of the tree. He then declared that he could not bless the tree that it might have good fruit, nearby water, beautiful boughs, fine wood, because it had those things already, so he blessed it that its saplings might be blessed just as it was blessed. Likewise Abraham came into a spiritual wilderness, and was blessed with fine descendants. The ten days make a further connexion with the story just mentioned concerning one of Abraham's ten trials. In this place the parable is ascribed to R. Levi, but in our Arabic source it is ascribed to R. José. There are several sages of this name, but this is probably R. José b. Halafta, who was noted for his piety. In particular, R. José took a wife in levirate marriage, which is required to be done not for personal gratification, but purely to fulfil the commandment (Deuteronomy 25.5.) It is said of R. José that he had sex through a sheet, in order to reduce his gratification, and only had sex enough to father half a dozen sons, all of whom became great scholars, thus illustrating the truth of the parable, for he was like the tree blessed in its qualities and producing good saplings.
This story is taken almost verbatim from the Babylonian Talmud, Ta`anit, 24b. Hanina was a Mishna teacher of the first century C.E. who was noted for his great piety. Even his donkey refused to eat untithed produce. His prayers were routinely answered. One day while traveling, he prayed for the rain to stop, and it stopped. When he got home, he prayed for it to resume, and it resumed.
R. Simeon b. Yohai (mid-second century C.E.) was one of the most famous Jewish sages, and the reputed author of the kabbalistic classic The Zohar. In this story, he recommends his son to seek a blessing from two of his outstanding students. The son is devastated by an ambiguous blessing which on balance seems to be negative, but his father explains that he has misunderstood the words and interprets them appropriately. This story is certainly Talmudic, but I do not know the precise source.