After losing 3-2 on aggregate to AEK Athens in the third qualifying round, Celtic will not take part in the Champions League for the first time under Rodgers' stewardship and there have been reports of a falling out between the manager and the board.
Following the 2-1 second-leg defeat in Greece, Rodgers said the club had to 'always guard against becoming complacent' in a move which is only likely to add fuel to the fire.
"You do that by adding to the squad," Rodgers told BBC Scotland.
"It's pretty obvious, you have to keep progressing and keep getting stronger.
"We gave them lots of problems and over two legs, we felt we were the better team but you have to defend well.
"Two moments have cost us. We did it against Livingston, Alashkert, Hearts earlier in the season and now we've done it here. Ultimately, that's not good enough."