When exactly does the background check happen?

I’m a current sophomore with a GPA that will probably end up at ~3.62 after this semester (low because I bombed first sem freshman year). However, if I recruit for normal BB SA roles (most of which I believe recruit in spring) and I get an offer, at what point will they request my transcript? If I’m reasonably confident I can get my 3.62 up to 3.65 after spring, can I get away with putting 3.7 on my resume? I go to a competitive target (Mich/UVA/Cornell). What should I do?