According to the prophecies in Old Testament, the Messiah must be a descendant of David. In the New Testament the apostle Paul starts his letter to the church in Rome:

“Jesus the Messiah, our Lord, who was descended from David according to the flesh.” (Romans 1:3)

In the past, nobody questioned that Jesus was a descendant of David, neither historians nor the Sages.
This was because the genealogical scrolls of the people of Israel were accessible in the Temple. If somebody disagreed he would have speedily pointed out the mistake referring to these scrolls.
Everyone knew that Jesus was indeed a descendant of David. If Jesus was not from the Davidic line, both the priests and the Rabbis during Jesus's time, not to mention the Talmud, would have pointed this out.

However, in the Talmud Jesus is referred to as somebody who was considered a descendant of David.

Had they known that it was not true they would have used the opportunity to point this out. However, that never happened. Rather, in the Talmud, Sanhedrin 43, page 1, it is said that Jesus was “close to the Kingdom.” They then explain that he was close to the Hasmonean dynasty.
And indeed, according to the NT documents, Jesus was not only close to the royal line but also to the priesthood. The NT says that Jesus was a descendant of David from both sides of his parents, both in regard to his biological mother's ancestry and of his adoptive father. In Judaism, an adoptive father was always considered father in every respect. Based upon this as well among the nations the notion of “Apotropos” evolved. In this matter, pay attention to this rabbinical commentary:

“On what basis do we relate Aaron's sons with Moses? Since he taught them Torah. And it is written about him as if he begot them. And therefore it is said that on the day that the Lord spoke to Moses at Sinai: Who made the sons of Aaron be called by Moses? The Torah that God spoke to Moses at Mount Sinai. Thus you should teach that whoever teaches his friend's son Torah the scriptures say that he has begotten him.” (Midrash Aggada, Numbers 3, A)

Simply put, the commentary states that Moses was the father of Aaron's sons only because he taught them Torah.
And next to that, the Jewish tradition itself states that the Messiah should not have a biological father. We already discussed this matter in detail in videos about the Son of God and the virgin birth of the Messiah.
However, we would like to quote Prof. Hananel Mak, Talmud Department University of Bar Ilan from his work on Rashi's Rabbi, Rav Moshe HaDarshan:

“The commentary is based on the combination of the human character of the Messiah who does not have a father of flesh and blood, the prophecy of the suffering and servile servant of Isaiah 53, which is the section “My servant will act wisely” and Psalm 110, that describes the relationship of God with the one sitting at his right and with Melchizedek.” (Prof. Hananel Mak)

Therefore, Prof. Hananel Mak acknowledges the fact that Rabbi Moshe HaDarshan, admired by Rashi, interpreted from Isaiah 53 that the Messiah would not have a biological father. 
With the destruction of the 2nd temple the genealogical scrolls were also destroyed. And therefore, in our days, there is no way to know someone's exact ancestry.

This is a lethal blow to the current rabbinical tradition, because if God gave the people of Israel this sign to identify the lineage of the Messiah, we have been unable to know for sure if someone is from the line of David or not since the time of the temple, which was destroyed 1900 years ago.

Moreover, our people's lineage has been become intermingled over the centuries, both on a national and international level. That's why there are Jews with dark skin, Jews with blond hair, Jews with slanted eyes, middle eastern Jews that look like other people groups from North Africa, Jews with blue eyes and European skin color, and so on. Over the 2000 years of exile all our lineages as Jews were mixed.
But today, centuries after the destruction of the temple and all our the genealogical scrolls, the Rabbis keep trying to restart the discussion by claiming that Jesus is not from David's line. See an example from Rabbi Michael Skobac who claims as follows:

“It turns out that Yeshu(a)'s ancestry on Joseph's side associated to King David, goes through a King with the name Jehoiachin. The problem is, that in Jeremiah chapter 22 this king is being cursed by God. ‘Thus says the Lord: Write this man down as childless, a man who shall not succeed in his days, for none of his offspring shall succeed in sitting on the throne of David and ruling again in Judah.' From this passage in Jeremiah we understand that every descendant of Jehoiachin is disqualified from being the Messiah and therefore Yeshu(a) is disqualified.” (Rabbi Michael Skobac)

And indeed, the Rabbi is right. King Coniah, known also as Jehoiachin, was cursed. But what the Rabbi forgot to tell you, or might not know himself, is that in the book Haggai, chapter 2, verse 23, Zerubbabel, Jehoiachin's grandson, reigns over the tribe of Judah. And at the end of the chapter God tells him:

“… [I will] make you like a signet ring, for I have chosen you…” (Haggai 2:23)

This means that though God cursed Jehoiachin, God later reversed the curse and his descendent, Zerubbabel, indeed reigned once again over Judah. Turn to Rabbi David Ben Yosef Kimhi's commentary on Jehoiachin. He too states that God forgave and withdrew the curse from Jehoiachin's lineage. One more example comes from the Rabbi Daniel Asor.

“We find that Yeshu(a)'s genealogy written in the gospels of the NT are mixed up. Matthew says that Yeshu(a)'s grandfather was Jakob. However, in Luke's book it says that Eli was his grandfather.” (Rabbi Daniel Asor)

Rabbi Asor claims that both Matthew and Luke present Yeshu(a)'s genealogy in their gospels. And that these genealogies contradict each other, since they mention totally different names. That's right! The genealogies are different from each other and indeed contain different names.
This should have helped the Rabbi understand that they do not present the same genealogy. Rather, Matthew presents the genealogy of Jesus's father, while Luke, presents the genealogy of Jesus's mother. These are two different genealogies and not one.
Today, we don't have the genealogical scrolls, therefore it is impossible to prove the lineage of any contenders claiming to be the Messiah since the time of the second temple. In Jesus' case, however, the scrolls were still available to been seen, and even the writings of his adversaries prove that he fulfilled the criteria. He was indeed from the line of David.