libssh2 can do the SSH handshaking and user authentication stages in non-blocking mode. I am not completely sure the same can be currently done with Net::SSH2 though.

The trick would be to call $ssh2->blocking(0) just after creating the Net::SSH2 object and then calling connect and auth_* while checking for LIBSSH2_ERROR_EAGAIN errors and then using the select function to wait for data to arrive at some of the underlaying sockets.

I have done something similar for the Net::SSH2 backend of my module Net::SSH::Any and would have to say that it is going to be a difficult task and would probably require fixing bugs in Net::SSH2 (in that case, report them through GitHub, I would be looking and could help).