You are able to place the trap, and if the creature steps on it they will be subjected to it's effects.
Most traps have triggers, and this one is explicitly specified, and that trigger is "A creature that steps on the plate."
The description of the hunting trap, in the PHB under adventuring gear, doesn't specify a creature's space or the square they're standing on. The description only states if they step on it. Assuming he saw you plant the trap, they wouldn't step in it, willingly, unless they were unintelligent. But maybe they could be pushed onto it, or another creature could step on it later.
A DM might rule that being on the same square counts as "stepping on the trap," but, I would rule that you can't place a trap under a conscious target, without triggering it yourself.
I would rule this like the following:
- You place the trap, using your action
- The DM decides whether or not your trap is hidden. This can be determined by deciding if you were seen setting the trap. If you were seen setting this trap, the trap isn't hidden. If the battle is hectic, he might rule that the trap is hidden.
- The DM then sets a DC to notice the trap, and checks that against the creature's passive perception score.
- If the creature doesn't notice the trap, and then moves (because you can't place it directly underneath the creature), the DM will determine whether the creature stepped on that trap or not, and then consult the trap description if the creature stepped on it.