Bill Belichick's cheerleader girlfriend, Jordon Hudson, is so flippin' mad over some recent jabs a radio host threw her alma mater's way ... she's now issuing a challenge to the woman in an attempt to completely embarrass her.

The beef all began on Monday ... when Winnie Akoury -- who appears on Kiss 108's "Billy & Lisa in the Morning" -- trashed Bridgewater State University's cheerleading program.

Akoury stated the school's team -- which 23-year-old Hudson competed on just a few years ago -- is "Division III -- which isn't even, like, competition level."

"I could have been a Bridgewater State cheerleader!" she continued.

Word got around to Hudson pretty quickly ... and on Wednesday, she made it clear as day she's furious over the barbs.

Hudson took to her Instagram to defend her former university -- before she dared Akoury to compete in the team's tryouts this winter.

"And, since you are SO confident that you could make the team, (as you stated LIVE on the air (twice)); I'd love for you to bring a @Kiss108 camera crew to film your undeniable domination of our pathetic little DIII cheerleading program. 🤗"

"P.S.," she added, "Kiss my.... National Championship Ring💍🤭."

Winnie went on to address Hudson's call-out on Thursday ... and she was adamant she might be able to take the challenge -- noting she squats a lot in the gym.

"I could be a part of that team that makes sense," she insisted.