Is there any free RSS reader client for Mac that can give me a notification if there are new posts to read?
If I understand your question correctly, NetNewsWire does this.
It's free, but ad-supported; you can pay extra to remove the ads.
Though it will eventually be a paid app in the Mac App Store, the Reeder beta is currently free. It's a very well-designed, highly polished app that's actually gotten me back into reading RSS feeds:
If you need a notification for new posts to read, you can configure Reeder to display a count of unread RSS items in the dock:
NewsFire has Growl integration.
Or you can use Google Chrome and the Google Reader Notifier.
Hopefully Google will build this in reader in the next update. Google Mail already has this build in.