Nick Young & Iggy -- Relationship On the Rocks ... Nick Trying to Save It

Nick Young is fiercely trying to save his relationship with Iggy Azalea ... with multiple sources telling us it's hanging by a thread.

We're told by people very close to the couple that both Iggy and Nick are pissed about the D'Angelo Russell video ... in which Nick apparently admits to cheating on Iggy over the summer.

Our sources say the engagement is in jeopardy -- and Nick and Iggy are "not on good terms."

Sources close to Nick tell us he's crushed -- telling people he's madly in love with Iggy and is focusing on trying to repair their relationship.

As for the video ... multiple sources tell us it was shot MONTHS ago.

We're told everyone involved believes the video was obtained and leaked by a 3rd party -- not Russell.

That being said, we're told the tension between D.R. and Young is thick -- but a Lakers honcho is planning to sit them down to hash this thing out man to man as soon as possible.

The Lakers want them to bury the hatchet so they can move forward.

And then there's this ... we're told Nick has already contacted his lawyers to weigh his legal options -- but it's unclear if he's planning to go after D'Angelo or whoever posted the video.