What is the Linux client of which you speak? Gog doesn't support Linux in that way. Are you talking about the blue dot on the web site, indicating updates? That doesn't work most of the time. Are you talking about lgogdownloader? I suppose you can set up a cron job to look for updates with it.

I mean the Linux client of the game. I wondered if the game, when run, would give a notification that there was a newer version available.

I see. Well, I don't let any game touch the 'net, so I can't help there. I've just never heard of the game executable being called a client, except for multi-player on-line games.

I wouldn't be a big fan of the game going to the internet myself due to privacy concerns. But maybe try using lgogdownloader, it will help you download updates in an automated way. I haven't tried the newer galaxy options but it should even be able to handle the installation with them.