I'm not Jewish or an expert in Jewish genetic genealogy. However I am a haplogroup administrator of J-M241 and J-M172 which do contain some predominantly Jewish lineages, so I have some experience with these lineages.
Genetics does not equal a religion, in the same way it does not equal a language or ethnicity. These things can change from one generation to another quite easily.
However there are three lineages within J-M241 that appear to have, for the most part, passed down the Jewish tradition along with their Y chromosome for several hundreds of years.
All men in this lineage descend from a man who lived most likely in Portugal, though Italy is a second possibility, about 1200 BC years ago according to YFull estimate. At least two of this man's lineages are today Jewish, which makes the ancestor himself a possible if not likely candidate for having been Jewish himself. Of course 1200 BC is too early for a Jew to have been living in Portugal, I will discuss this later.
One of the lineages, J-Y36166, claims descent from Yahia Ben Yahi III, also known as "El Negro", a Sephardi Jew who was advisor to Alfonso I of Portugal.
The tradition is that his father, Yahia Ben Rabbi, descends from the Exilarchs of Babylon. However, given that no near sibling or cousin lineages of L283 are found in the Middle East, and that ancient DNA of a related lineage was found in Croatia 1600 BC, it is more likely that the ancestor of these lineages was already living in Europe, and Portugal is the best guess based on 3 of 5 separate lineages being found in Portugal today.
Note that if the YFull estimate that the most recent common ancestor of all CTS6190 lived in 1200 BC is correct, then this man couldn't have been a Jew as Judaism did not yet exist in Europe. So either this estimated date is wrong or at least two lineages of CTS6190 independently converted to Judaism at a later date. The early estimate of CTS6190 TMRCA does not reconcile the lack of geographic support for an origin in the Middle East.
The men in this lineage descend from a man who lived 800 AD. Almost all have a history of Ashkenazi tradition. It is not yet clear where the ancestor of this lineage, Z8429, was living in 600 BC, as he sired geographically diverse siblings Y12007 (England, Germany, Czech, Russia, Tatar, Chechen), CTS11760 (England, Spain, Italy, Germany, Poland, Slovakia), and additional Norwegian and English lineages.
Z39653 appears to have originated in Eastern Europe though research is still ongoing. Because none of the siblings from 600 BC have a Jewish tradition it makes more sense that this ancestor was not a Jew and that at some later time, a man in the Z39653 bottleneck from 600 BC to 800 AD converted somewhere in Eastern Europe, perhaps Ukraine or Poland.
A third cluster is represented by a single man in the J-M241 project. He traces descent to Istanbul, Turkey and his paternal line is of Romaniote Jewish origin.
On the YFull tree he is YF14253. He's negative for Z615 meaning his ancestors parted ways from the ancestors of most J-L283 before 3400 BC. Z585 and Z600 are no-calls but Z627 is positive.
On May 17, 2019 I contacted the Avotaynu Project on FTDNA to join four kits they manage, which are 12 markers matches to our Romaniote, to the J-M241 project. They have not responded or joined the project so J-M241 research into this interesting basal lineage is at a standstill.