The LAPD is looking into allegations that James Harden roughed up a photog in L.A. on Tuesday night -- allegedly breaking the guy's finger in the process ... TMZ Sports has learned.

Law enforcement sources tell us ... the accuser told police he spotted Harden in front of a lounge in Hollywood and tried to ask him for his opinion on Donald Trump. 

The photog -- not a TMZ guy -- claims Harden became aggressive and pushed the camera. The photog claims his finger was caught in the cam ... and was broken in the skirmish. 

The photog filed a police report ... and cops are now investigating to see if there's evidence to back up his claims -- including medical records and possible video. 

Important to note Harden has NOT been charged with a crime.

We reached out to Harden's camp -- so far no word back.