Rabbi Tzvi Hirsch Kalischer was the first to coin the phrase: Zera Yisrael
Rabbi Kalischer used this in reference to someone born to a non-Jewish mother but had a jewish father. He referred to them as Zera Kadosh - Holy Lineage.
However, Rabbi Kalischer did not use it in a halachic context.
The most vocal and bold Halachist to apply this concept was the Former Sephardic Chief Rabbi of the State of Israel Rav Ben Zion Meir Chai Uziel (שו”ת משפטי עוזיאל, כרך ב יו”ד סי’ נב) who argued that the concept of Zera Yisrael is considered so legitimate that it can be applied to determine lineage. For example, if the non Jew -zera Yisrael father is a Kohen, despite his mother being a gentile, once he converts his "zera Yisrael" kicks in and he is considered a full fledged Kohen and can partake in the many benefits/ prohibitions that apply to Kohanim.
Rabbi Uziel was deeply concerned about the fate of children born to a
Jewish father and a non-Jewish mother. Such children, although of
Jewish stock (zera Yisrael), are in fact not halakhically Jewish.
Children raised in such intermarriages will be lost to the Jewish
people entirely. Thus, it is obligatory for rabbis to convert the
non-Jewish mother in order to keep the children in the Jewish fold.
Rabbi Uziel noted: “And I fear that if we push them [the children]
away completely by not accepting their parents for conversion, we
shall be brought to judgment and they shall say to us: ‘You did not
bring back those who were driven away, and those who were lost you did
not seek.’ (Yehezkel 34:4).”
Some (source for quote above as well) have argued that the concept of Zera Yisrael is also used as a justification to streamline the conversion process. Where traditionally a certain amount of time is given in order to bring the prospective convert up to speed in torah knowledge, or acclimation to an observance life, as well as to discern their true motivation for conversion. Some have argued that in the case of a Zera yisrael, we speed up the process and accept them right away as it is a mitzvah to bring that back under The Wings of the Schechina.
However, this concept (if legitimate) is only limited to those that are literal descendants of Jewish lineage.