Sounds like it's coming from a Dell Optiplex 3010.
Optiplex 3010 Technical Guidebook
Pg. 10 shows the max processor as the I5-3450. The I5-3570K is also a 77 watt processor, so I'm not sure why it's not listed here. It might work, but I'd want confirmation before making the purchase.
If it's not listed as a supported processor even though it's from the same gen, it's usually because the processor was released after the motherboard was released. Whether it'll work will then depend on if there was a BIOS update which supports the new processor. The newer processors are not carbon copy clones of the older ones, just clocked differently. Intel makes a few small tweaks and bug fixes. The motherboard has to know about these slight differences in order for the newer processor to work even if the architecture is the same as the older processors. The BIOS update does that (adds support for newer processors).
You can either spend an evening reading through the changelog for the BIOS updates for this motherboard, looking to see if support for the new CPU is mentioned. Or since this CPU is so old and relatively cheap, you can just buy it and cross your fingers hoping it works. Either way, make sure you update to the latest BIOS before switching processors. Updating the BIOS requires a functioning CPU. So if your current BIOS doesn't support the new CPU and you install it, you will need to remove the new CPU and reinstall the old CPU, update the BIOS to a newer one which supports the new CPU, then remove the old CPU and install the new CPU.