Can I update Sierra on my 2013 iMac?

Which OS you can upgrade to will be governed by which model and year your mac is.

An iMac from 2013 can upgrade to either macOS High Sierra 10.13.6, macOS Mojave 10.14.6,

or macOS Catalina 10.15.7.

As your mac can upgrade to the 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, Catalina,

Open Safari and click on this link, 

How to get older versions of macOS

Go to the Use Safari to download macOS 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 tit 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 free 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.

The Age of this computer will Govern what newer version of macOS it will Qualify to install. Use About this Mac from the Apple Icon on Desktop - the required Information will be there.

With the above information in-hand - follow this How to get old versions of macOS and verify what version this computer Qualifies to install.  For Best results use Safari to commence the download as Others may not work.

Safari will open a redirect and open the Apple Store where the download can commence

One may consider upGrading to High Sierra first if considering upgrading to any version above High Sierra. This may better position the computer for the Drive Conversion to the newer APFS format as well as any EFI BOOT ROM required to use the newer file format.

Tips Big Sir Upgrade

Make Time Machine Backup  is very useful and can be used to Revert to Previous Working macOS

Notation - Mojave is the Last version to support 32 bit applications / extensions and Drivers. Catalina, Big Sur & Monterey supports only 64 bit and there is no workaround.

Suggest using Go64 to find 32 bit application before attempting the upGrade. Some or many of the Application already installed may required update or upGrades to 64 Bit Versions.

Are there any AntiVirus, Disk Cleaner, Optimizers, Un-installers, etc installed which should be removed as per Developers Instructions. They are useless, unneeded, cause havoc and interfere with the normal operation of the OS and may even Corrupt the OS requiring a Reinstallation