Finkel is the son of Rabbi Chaim Zev Finkel, from whom he inherited the position of Mir "mashgiach", and the grandson of Rabbi Eliezer Yehuda Finkel, who led the yeshiva in Poland and brought it to Israel.
Finkel is the son of Rabbi Chaim Zev Finkel (1906--1965), founder of Yeshivas Heichal HaTorah in Tel Aviv and "mashgiach" at the Mir in Jerusalem. He is the grandson of Rabbi Eliezer Yehuda Finkel, who led the yeshiva in Poland and brought it to Israel.