Can I updates my 2010 Mac Pro machine with OS X Version 10.11?

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.

A 2010 Mac Pro can upgrade to macOS High Sierra and with a 'Metal' compatible graphics card

can upgrade to macOS Mojave. .

For older El Capitan, High Sierra and Mojave,

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.

Read the information for Mojave and Mac Pros.

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'.

Back up your data, check your applications for compatibility, and depending on what you're trying to do, click on El Capitan or High Sierra.

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