We've called around Ann Arbor ... the first round of rushing began Tuesday, and multiple sorority members tell us Barack Obama's youngest daughter wants no part of it.

For those unfamiliar ... rushing is the process of exploring all of a university's sororities, and interested students at Michigan must register before a deadline. We're told Sasha did not register.

TMZ broke the story ... at the beginning of last semester, UM sororities were champing at the bit to recruit Sasha in the spring, and she was dubbed a "rush crush."

All signs were pointing to Sasha rushing and pledging Alpha Chi Omega (AXO) -- she stayed at AXO's house once before enrolling, and her close friends from high school are members there -- but now she's following her mother's and sister's footsteps by not rushing.

Michelle Obama didn't join a sorority at Princeton, and Malia isn't living the Greek life at Harvard, where sororities are not recognized by the Ivy League school.

Looks like Sasha is looking for a similar college experience.