Unable to update late 2015 iMac to macOS Big Sur

In addition to @John Galt's advice & information provided by @Servant of Cats........

Personally I would recommend downloading the macOS 10.11 El Capitan installer (@Servant of Cats provided a link to download the macOS installers direct from Apple).....any computer & OS which can access the Internet can be used to download the El Capitan installer DMG file, but you will need to transfer it to the iMac where you can run the .pkg installer/extractor included in the DMG file to extract the real installer into the Applications folder. Then you can create a bootable macOS 10.11 El Capitan USB installer using the instructions in the following Apple article:

Create a bootable installer for macOS - Apple Support

I guess you can try downloading the macOS 12.x Monterey installer since that would be downloaded through the Apple App Store instead of a web browser (would cut down on the work needed assuming you can download stuff from the App Store on that iMac).

I'm recommending creating a bootable macOS USB installer if at all possible before attempting to erase the iMac just in case you may encounter issues with Internet Recovery Mode (Command + Option + R). Otherwise you will need access to another working Mac from about 2007 to mid-2021 to be able to download & create a bootable macOS USB installer (macOS 10.11 to 12.x).

Skenny16 wrote:

I have just been given an old imac 21.5 inch (late 2015) from family member. Still

works

so want to use it whilst I can. But to run a few program and I need to update the OS. I’ve looked into and it will only

go up Big Sur 11

If this is an iMac (21.5-inch, Late 2015) or iMac (Retina 4K, 21.5-inch, Late 2015), it can actually go up to Monterey. Upgrading to Monterey might not help a lot in terms of applications availability, and if that Mac is starting up off of a mechanical hard disk, you might want to let sleeping dogs lie – but Monterey would be an option.

How to download and install macOS - Apple Support

Which I’m think because the version of the Mac apple does not have support for it so can’t access.

Whatever is going on, it's not that. Big Sur's version of Safari and even earlier versions work fine to access Apple.com and nearly every other website there is — "nearly every" since I am not aware of any that don't work.

Have that family member comply with What to do before you sell, give away, trade in, or recycle your Mac - Apple Support. Those instructions are not optional and must be complied with lest you continue to experience problems using that Mac.

Big Sur no longer gets the latest versions of Safari – but Big Sur's version of Safari should be sufficient to access the Apple Web site.

There are several third-party Web browsers whose current versions will run on Big Sur. These include

• Firefox
• Brave
• Opera

Once you get your family member to prepare the iMac for a new owner (you), you may want to consider installing one of these browsers, to have in addition to Safari.

Likewise, while Microsoft doesn't support new installations of Office or Microsoft 365 on Big Sur, current versions of LibreOffice will run on Big Sur.

Comply with as much as What to do before you sell, give away, trade in, or recycle your Mac - Apple Support as you are able.