It is not mentioned at all in the Gemara, and it is based on teachings of the Arizal and the Zohar haKadosh.
Those kind of things are not usually practiced in yeshivos.
Mashgichim do talk about these days as a time of teshuvah and tikunim, but fasting and selichos etc. are not things yeshivos encourage since it makes it more difficult to learn Torah which is the most important thing in Judaism.