is there a free word-processing software for iMac

There are many. However what comes to mind first is Apple's Pages app is free and comes with every new Mac. If we knew more about your system then we could also suggest something more specific. We need the year and version of Mac OS installed.

As far as web based word processors you can find many including:

• Google Docs. Google Docs. ...
• Zoho Writer. Zoho Writer. ...
• ONLYOFFICE Personal. ONLYOFFICE Personal. ...
• Microsoft Word Online. Microsoft Word Online. ...
• Live Writer. ...
• Shutterborg. ...
• Writer.

by using Google and search

In addition to rkaufmann's recommendation for Pages (which converts MS Word Documents) and the others he lists, for basic word processing there is TextEdit. TextEdit is my go-to application for any basic writing needs.

Tphputs wrote:

So, do you think there is a 'Pages' software version on my iMac from 2010?

El Capitan ver 10.11.6

Yes, either on your Mac or on the OEM discs that shipped with your Mac. But now Pages is free in the AppStore and if you upgrade to Sierra you can get the latest version.

Tphputs wrote:

So, do you think there is a 'Pages' software version on my iMac from 2010?

El Capitan ver 10.11.6

No, beginning in 2013 it was bundled for free. You can get it for $19.99 from the Mac App store, after you buy it updates are free after that.

I was able to find this article although I can't say from personal experience that it works or not but it's worth a try.

https://www.cultofmac.com/306118/download-apples-iwork-apps-older-macs-free/

http://www.libreoffice.org/ - free office suite (not an online service).

Hmm... I was pretty sure iWork came on my OEM discs when I bought my 2010 iMac. I know I've had it forever and I didn't pay. I'll have to check the discs, maybe I ported it over from an earlier Mac.

Its so long ago I had to look up that article so who knows for sure. The OP can try the article's suggestion so it's up to him to find out.