As @olasd pointed out, async is a keyword so let's not use it as module name :-)

My concrete proposal is then to add a new swh.web.client.aclient (a for async) module, which will be a sibling of the current swh.web.client.client.
aclient will reimplement and expose the current public client API, all as coroutines.
client will maintain the current API, but all implementations will be asyncio.run() wrappers.

Note that this means that we will change the HTTP client implementation of the entire module, no matter which sync/async API is used, from requests to aiohttp. (Because there is no async support in requests AFAICT.) This is fine by me, but we should all be aware of this (cc: @anlambert for info).
It seems better than having two different implementations, one using requests for the sync part and one using aiohttp for the async one.

(As discussed with @seirl I'm not considering the option of keep on using requests and run it in a dedicated thread, because starting threads behind the back of client code seems like a bad design pattern.)