NBC and FOX are going full bore against each other to bring "American Idol" back. 

We're told both networks have submitted bids to FremantleMedia -- which owns the show -- and a decision could come as early as this week.

Multiple sources familiar with the situation tell TMZ ... NBC wants the show in part to resolve a problem with "The Voice." We're told the network doesn't want to oversaturate the market with the show, which airs twice a year. We're also told Adam Levine only wants to do 1 cycle a year.

If NBC gets 'Idol,' it would rotate 'A.I.' with "The Voice" and each would air once a year.

We're also told NBC has Ryan Seacrest as the favored host the second time around, and FOX would also entertain bringing Ryan back.

Our sources say the target date for bringing 'Idol' back is either Summer 2018 or mid-season 2019.

FOX never gave up on the show ... it just became way too expensive to produce with the talent costs ... Ryan, J Lo, Keith Urban and Harry Connick Jr. The reboot, we're told, would have a new panel of judges.