Issue seems to be solved in :
https://issuetracker.unity3d.com/issues/linux-roslyn-fails-if-there-is-a-newer-libssl-than-1-dot-0 (duplicate)

It isn't only Unity Package Manager, also for whole Unity

Got the same problem.
I don't have the package manager, and i got logs errors without any description (only datetime is written).

Just as jasonboukheir said, i got "No usable version of the libssl was found" in the logs.

This issue is critic as the game can't be launched / built with these errors. Please look into it :)

Version :
Ubuntu 19.04
2019.1.0f2 (Also present in the alpha 2019.2.013a)

In the editor log, I'm getting "No usable version of the libssl was found".

My openssl version:
1.1.1b

My libssl versions:
libssl-dev/disco,now 1.1.1b-1ubuntu2 amd64 [installed,automatic]
libssl-dev/disco 1.1.1b-1ubuntu2 i386
libssl-doc/disco,disco 1.1.1b-1ubuntu2 all
libssl-ocaml-dev/disco 0.5.5-1 amd64
libssl-ocaml-dev/disco 0.5.5-1 i386
libssl-ocaml/disco 0.5.5-1 amd64
libssl-ocaml/disco 0.5.5-1 i386
libssl-utils-clojure/disco,disco 0.8.3-2 all
libssl1.1/disco,now 1.1.1b-1ubuntu2 amd64 [installed,automatic]
libssl1.1/disco,now 1.1.1b-1ubuntu2 i386 [installed,automatic]

There seems to be a work around. If you have an existing project that was created on a Windows machine and you open that project on Linux the Project Manager will be usable from the Window tab but certain packages will cause unknown errors, namely TextMeshPro and ShaderGraph

TextMeshPro is also missing in the window. It may be a related issue there.