Discord is developing positional audio.
To get in on the action early, see this.

This issue is a placeholder to not forget this until they get that far.

It seems like it will not be possible to use Discord Gamebridge for this in the foreseeable future:
https://www.reddit.com/r/discordapp/comments/7t0jyl/gamebridge_dead_or_when/

However it may be possible to use Rich Presence to directly set the volume and pan of other users in the same audio channel.
https://news.ycombinator.com/item?id=13981914
https://discordapp.com/developers/docs/topics/rpc#setuservoicesettings

How would we determine which Discord user-id belongs to which MC client? Probably a server-sided mod.

Apparently this volume and panning isn't on a per-client-basis, but rather affects a user's volume and panning for everyone in the voice channel. :/
Source: Asie

It won't happen:

From Sept 29. 2019; Discord GameSDK Help; #game-sdk

<@53908232506183680> any info on Positional Audio being reintroduced?
@DutchAndDangerous#2296

<@225595810115485697> no, the extent to which we support positional audio is with the pan and volume abilities. As a general philosophy, we don't want our voice chat to be like Vivox. With a solution like Vivox, you can do positional audio, filters to sound like you're in a cave, underwater, etc. If that's what you absolutely need, something like Vivox would be your best choice.

We aim to make sure that Discord audio quality is as amazing as it can be, so we don't want a lot of mutating with the sound outside of pan and audio.
@mason#1337