can you switch a 2010-2011 iMac's cpu?

Changing CPUs would be a waste of time and money.

Back in the PowerPC days when CPU changes were a bit less of a challenge I found that I needed a 40% increase in processor speed to even begin to "feel" the CPU upgrade in actual use. That much processor speed increase may not be possible.

What slows most older iMacs is not the CPU ** their mech hard drives. Here is where we need an exact year to best advise you. The 2011 iMac had a drive bus that was twice as fast as the 2010 and older models. An internal SATA 6G SSD in a 2011 iMac can do 500MB/sec transfers all day and the computer would feel rather nimble ever with the original processor.

The fastest mech drive options in a 27-inch 2011 iMac could only do about 150MB/sec at best.

However, the drive bus in a 2010 iMac was only half that of the 2011 so adding an SSD would not be a big change in speed. Probably not worth the time and effort.

⚠️ NOTE: Because 2011 and older iMacs lack USB-3 ports, the oft-recommended external USB SSD solution is not an option for you. That could actually reduce current performance on slow USB-2 ports.

Here is a useful proof statement. The 2011 iMac 27 had three CPU options: 2.7Gz i5; 3.1 Gz i5; and 3.4 Gz i7. These are benchmarks scores for all three from the Mactracker database: note the minimal increase in going to the faster and compatible CPU, the i7:

Yes the i7 is exactly 40% more but, as I said, at the keyboard that would barely be noticeable. The computer won't load boot volumes or apps any faster.

I'm sure that the official answer would be "No."

Unofficially, I don't know if Apple socketed the CPUs – and, if they did, whether you could still get Intel CPUs that fit the type of sockets that Intel used back then. I'm pretty sure that Intel will have changed their sockets several times since then, so that there's no chance that a modern Intel CPU would fit.

The CPUs that MacTracker says are in those machines are so old that you have to go to the

Legacy Intel® Core™ Processors

page on the Intel site to find their specifications.

Anyway, if you replace the defective hard disk, and the machine has at least 8 GB of RAM, I believe you will find that software availability – not CPU speed – is the limiting issue. You can't upgrade those Macs past High Sierra. Adobe and Microsoft don't support High Sierra, and even many developers who traditionally have supported older versions of macOS have moved on, or are starting to move on.