The App Store Updates page only informs you of updates available for the OS you are running.
Installing a newer OS is called an upgrade.
An iMac from 2013 can upgrade any available OS up to macOS Catalina 10.15.7
AS your mac can upgrade to the newer OS's like macOS Mojave 10.14, macOS Catalina 10.15,
I would recommend you upgrade to macOS High Sierra 10.13 first.
For High Sierra, Mojave or Catalina,
Open Safari and click on this link,
How to get old versions of macOS – Apple Support
Go to Check Compatibility to see if your mac qualifies for the OS upgrade you want.
Then go to the Download OS section and click on the OS you want to install.
Before upgrading make a full back up of your Mac.
Check that third party apps you use have updates available for the new OS.
https://roaringapps.com
Check that your printers/ scanners etc have updated drivers and software available for the new OS.
Connect your mac to your router via cable rather than rely on WiFi.
Do not use your mac while it is downloading
and do not use any other devices that share the same internet connection.
If the App Store says ‘Open’ then it is detecting a previous download of the install app
go to the Applications folder locate Install macOS ******.app.
Move it to the Trash, empty the Trash.
Restart, click on the link again, the App Store should say ‘Get' or 'Download'.
If you are considering upgrading to Catalina be warned it does not support 32bit apps.
Your Mac is probably running 32bit apps and these will need to be updated to 64bit apps,
some updates may be free, some may be paid. Use this app to discover which apps you have
are 32bit, https://www.stclairsoft.com/Go64/ then contact the developers of the software to see
if they are offering 64bit updates.
