'American Idol' Honchos -- We Could've SAVED Jermaine Jones ... If He Didn't LIE

The bigwigs at "American Idol" tell TMZ ... they would have done everything in their power to help Jermaine Jones clear up his legal situation ... if only he would have been HONEST about his past.

"A.I." executive producers Nigel Lythgoe and Ken Warwick called in to "TMZ Live" moments ago ... and explained why they HAD to confront Jermaine when they discovered he LIED about his criminal history. They said they were particularly worried when they found out he had used false names with police.

The "Idol" honchos explained, "There might be other false names and other charges out there that we just dont know about."

Lythgoe and Warwick tell us lots of kids that appear on the show are entangled in legal situations ... but the producers insist, "We help them where we can in clearing them up."

The producers say they captured the confrontation on tape ... and say he was "very apologetic."