Yes, XP will run on an i3. And now for the caveats:
a) You will need the XP installation media and a valid Product ID for it. Which means that you will need a retail copy of XP. Transferring a Product ID from an OEM version of XP (that is, from an XP that came with another machine you or someone else might have purchased) is not legal, such licenses are valid only on the hardware that came with XP.
b) If your machine has more that 4GB or RAM, you'll need the 64-bit version of XP to make use of all the memory. You can run the 32-bit version, but you will see only about 3.5GB of the RAM.
c) You will have to do a search as to whether there are XP drivers for your hardware. In general, XP will run fine with the i3, your hard drive (assuming it has a fairly standard serial IDE controller), your hard-line network card, and basic video functionality (that is, 640x480 and 800x600 screen resolution). Any addition capabilities (wifi card, full video capability, etc) will require drivers. Fortunately, you say you have a desktop, so the parts might be somewhat standard and XP drivers might be available for them. XP Windows Update might even find a few such drivers, though getting the drivers from the manufacturer's web site would be better.
Having said all of that, my recommendation is to stick with the OS you have (probably Win 7), install a VM server, such as VirtualBox, and create an XP virtual machine.