The i7 870 has TDP of 95W but thats irrelevant the power consumption is of 121W and the 21.5 imac doesn't have the juice for that.
But ! the i7 870S (wich is the low-power variant) has the TDP of 82W and squeezes 66.63W from the power source. Bingo ! We have a winner.
I can confirm it works, i have the 21.5 mid-2010 imac with the i3 550 (dual core) now housing the quad i7 870S (2.66GHz turbo boost up to 3.6GHz, 8MB L3 cache). 4K video was a no go with i3 now plays smooth on i7, everything is blazing fast and I haven't even upgraded to SSD yet. the only thing I did notice is that in System Report I had 5.86GT/s Processor Interconnect Speed now its 4.8GT/s.
Did a lap on cpu before mounting in socket, reports 41 degrees Celsius on all cores never went past 60 in full load. The i3 had 44-46 on average use and difference of 1-1.5 degrees between cores.