Looks like it can work, RAW
The duration of Clone is listed as "Instantaneous", which leans us toward saying No, Soul Cage can't work on someone with a Clone prepared. However, Soul Cage is a reaction you take to a predetermined set of circumstances that you have prepared for--not unlike Counterspell. The triggers for Soul Cage may be harder to spot (final moment of death as opposed to a person making the motions of casting a spell), but an observant and prepared caster might still catch it.
Since Counterspell can counter other spells with a duration of Instantaneous, I would say the similarly reaction-cast Soul Cage would work the same way, in that the caster noticed a particular trigger and got the spell off at the last split second and Soul Cage grabs the soul before the transfer can complete.
Assuming Soul Cage does work, the soul would finish the transfer to the clone as soon as Soul Cage wears off.
I went and looked up the spells in question (PHB p. 222, XGE p. 165). Since Spells Do What They Say They Do, the Clone spell sets up four prerequisites for the soul to transfer to the prepared clone: the clone is mature, the original body is dead, the soul is free, and the soul is willing. There is no time limit given between "original body is dead" and the soul transferring for the spell to work successfully, and the existence of the "soul is free" stipulation means there are cases where a soul could be trapped upon death instead of just automatically transferring.
Also, note that Soul Cage, as written, just releases the soul when you are done with it, no text about using up the soul or where it's released to.
So, if you've successfully snatched a soul via Soul Cage and the spell has since ended, the soul is just released freely. Now, all four conditions of the Clone spell are true, and the soul can complete the transfer.
Still talk to your DM about it to see if you can do it. Not all DMs go by RAW all the time, and in the process of writing this answer I came up with three or four ways a DM ruling or houserule would totally alter the interaction of spells and soul here.