"American Idol" has completed the arduous and super expensive process of snagging 3 judges for the show's reboot ... and it didn't come a day too soon.

Sources connected with the production tell TMZ, ABC and Fremantle, which co-owns the show with Core Media Group, will announce as early as Friday night they have signed Luke Bryan and Lionel Richie as judges #2 and 3.

Luke and Lionel will join Katy Perry ... and join her fast. We're told shooting begins Tuesday.

We're told although Luke and Lionel have agreed to join, there could be a delay in announcing if the lawyers start nitpicking.

Production sources tell us, the show has gone way over the intended budget ... largely because Katy Perry held them up for $25 million. Ryan Seacrest's making somewhere between $10 and $15 mil, and that pretty much blew the budget sky high.

We're told Luke signed for around $13 million and Lionel got close to $10 million. It's going to be difficult to turn a profit unless the show is an out-of-the-park home run.