Can a 2013 iMac's drive be replaced to run new software?

Replacing the drive will not let you run a more recent version of macOS. Catalina is as far as any 2013 iMac can go.

Catalina is enough to run

• The current version of the Firefox Web browser
• The parts of Affinity Studio that are "free with registration." (The AI stuff that requires a paid subscription requires an Apple-Silicon-based Mac running Ventura or later.)
• LibreOffice 25.8.4. The current version (26.2.0) requires Big Sur or later, but 25.8.4 is compatible with Catalina. Right now, there is a link to 25.8.4 on the main Download page; this may eventually move off to the archive.

It is not enough to run

• Current versions of Safari, Pages, Numbers, Keynote, and GarageBand
• Current versions of most major third-party browsers other than Firefox
• Applications from vendors like Adobe and Microsoft who have a policy of supporting only the "most recent three"

If you're referring to the OS, no.

If you're referring to something else, only if its system requirements specify a higher-capacity drive; it's easier to move content to an external one instead.

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Timsgreenapple wrote:

I have a 2013 iMac Catalina can I replace the drive to have it run current updated software?

Nope' Catalina 10.15 is the highest macOS that a 2013 iMac can run.

Click the apple logo top left of your Menu bar and select “about this mac” to get exact details; Then, use the following link and go to the section called “Check macOS compatibility“ against your machines info.

Find out which macOS your Mac is using - Apple Support (CA)

You can update it to the latest possible OS as per that link, And you can find the operating systems in the app store to download.

Timsgreenapple wrote:

I’m asking because FaceTime app will not connect to FaceTime calls. I can send messages and make a regular calls just not FaceTime. It says call Failed. Any suggestions?

According to Continuity features and requirements for Apple devices - Apple Support, iPhone Cellular Calls only requires OS X Yosemite 10.10 or later. The feature has been more reliable on my current Mac – running Ventura and Sequoia – than it was on my previous one, which was running High Sierra.

Make and receive phone calls on Mac, iPad, and Apple Vision Pro - Apple Support

FaceTime calls that go over the Internet (rather than through the cellular network, via an iPhone) only require you to be running OS X Mavericks 10.9.2 or later.

FaceTime requirements on Mac - Apple Support

You will want to update Catalina to the just-released 10.5.8 by January 2027.

TidBITS – Apple’s Certificate-Extension Updates Continue for Older Operating Systems

If you have to consider getting a new Mac consider the following: a 10 Core Mac Mini M4 with 16 GB RAM and 1 TB SSD ($1084 US) with a 32" LG 4k monitor ($331 US) from Amazon.com)  is $1415 US which is $541US less than a similarly configured new 24" iMac ($1956 US)  re 07/2025. Keep your keyboard and mouse from your current Mac for use with the Mini. 

I got the above Mini with a 32" LG 4K monitor w/speakers for only $60 US more than the 27" (didn't know the Mini had a speaker).  There are many monitors available in 27" size for $100 US to $500 US depending on what features you want.  The monitor prices are before tariffs.

Just some food for thought.