Upgrade Graphics Card on 27" iMac

Upgrading a GPU in an iMac at best is not a trivial task and in most cases is not possible for a number of reasons. First is the hardware on Macs is designed to run together, Apple designs and builds computers to run well extremely well with the hardware and software they provide. Varying from that formula normally does not help. In addition, the hardware from year to year varies. So an iMac from 2010 likely won't be able to even fit a GPU from 2009 or earlier and 2011 and later.

My suggestion is before you go any further, evaluate what your needs are. If you have a need for a more powerful GPU then get a replacement machine. However remember if you decide to go the Hackintosh route, you are on your own. You would be violating the Terms of Use of using the operating system and anything that occurs is solely your problem, in short expect no support in any fashion except from other hackers.

You can put in a AMD Radeon HD 6970M with either one or two gigabytes memory,

but the performance gains would be slim, for a lot of work and expense..

https://www.ifixit.com/Guide/Graphics+Card/9553

These cards are known to be buggy, Apple/AMD used lead free solder,

with a lower melting point.

Solution is getting the card reballed

(Google BGA reball),

this will add $50-$75 to the costs...

I would upgrade the processor befor the graphics card.

The i7-870 (95 watts Thermal Design Power) is not economically upgradable (i7-880),

but the i3-550, or i5-680 (73 watts TDP) can be upgraded to a 4 core i5-750S or i7-860s/870s (82 watts TDP)

The i5-760 (95 watts TDP) can use the i7-860/870/880 processor.

What processor / speed do you have now?

I5-680, i5-760, or i7-870?

(I don't think it's an i3-550, max memory of that iMac is 16 gigabytes)

Any upgrade would be expensive, and difficult,

for only minor gains in performance...