Timing conflict with debt card effect
From the Cities rule book (4): 


  Important: monetary losses are always resolved at the end of a turn, after players have played their cards, possibly having paid for trade and/or obtained money...


The player is able to participate in the project, but he/she would have to take debt afterwards. 

As for whether that player has to participate in Great Project if he/she was going to, I haven't found ruling on it. Player consensus is this being optional and that player may make decision on it after they discovered debt threat.