It would help to know how exactly it fails.
This old thread may have something about the official client and Lutris geforce_now_lutris_installer_has_been_updated, one comment says
Join the lutris discord server. There is a pinned message with an attachment of the file that you need.
This other old one GeForce NOW finally works - General Discussion - Lutris Forums gives hope but I read elsewhere that GFN stopped working on Linux some time ago.
I have no experience with neither, sorry I can’t help much.
This 5 years old thread on nvidia’s website asks for a native Linux version, …crickets
https://www.nvidia.com/en-us/geforce/forums/gfn-general-chat/20/303146/geforce-now-on-linux/ and a recent comment indicate Valve may be doing something about it.
I see there is also the unofficial gfn-electron but that is basically the same as using a Chromium based browser, pretty sure it has the same limitations on resolution and frame rate.
I hope to be wrong, but it really looks like the answer is wait for a while and use the web version.