I started Terence Donaldson's Paul and the Gentiles. In my reading today, Donaldson talks about Jewish beliefs about the relationship of Gentiles to the Torah. Donaldson goes into five points-of-view within Second Temple Judaism, and I read about the first three. Donaldson also refers to rabbinic opinions on some of these viewpoints.
The first viewpoint is that the Gentiles are headed towards destruction. Overall, representatives of this first view believe that Gentiles should keep the Torah, but they do not stress proselytism, and Donaldson thinks that they only say that Gentiles should keep the Torah to justify God's eschatological punishment of them for not doing so. Believers in this approach include Jubilees, 4 Ezra, Testament of Moses, Pseudo-Philo, and the Apocalypse of Abraham.
The second viewpoint is that Gentiles should convert to Judaism. There are nuances in this position. Some Jews believed that they should actively seek converts, whereas others simply accepted Gentiles into Israel when they desired to convert. Shammai and some other rabbis, however, were skeptical about the motives and the value of proselytes. Donaldson believes that those who passively accepted converts represented the majority position. He also refers to some of the sources representing the first viewpoint in his discussion of this second viewpoint---4 Ezra, Pseudo-Philo, 2 Baruch---perhaps because, technically-speaking, they did maintain that the only way Gentiles could be saved in the Life to Come was through conversion to Judaism. Another issue is how far a Gentile had to go to become a Jew. Donaldson acknowledges that there were some debates on the precise boundaries, and that there were Gentiles who (say) kept the Sabbath but were not circumcised. But Donaldson's position appears to be that, overall, circumcision was a requirement for Gentiles to become Jews.
That brings me to the third category: Natural Law Proselytes. These are Gentiles who do some Jewish things, such as worshiping the God of Israel alone, but they are not circumcised. In certain passages, the third Sibylline Oracle, Philo, and Joseph and Asenath stress the righteousness of Gentiles who worship the God of Israel alone and practice virtue, but they do not say that these Gentiles are circumcised or observe the Sabbath. Josephus tells of Ananias, who said that a Gentile did not need to be circumcised to enjoy a relationship with the God of Israel. Donaldson does not seem to think that Jews considered Natural Law Proselytes to be full members of Israel, but Donaldson says that some Jews discuss these Gentiles in terms of proselytism because that was the only vocabulary they had to describe this phenomenon.