romerrr,
Your G4400 is 54 Watts TDP, while Intel's specified stock cooler E97379 (PCG 2015C) is 73 Watts TDP.
The 6700K is 91 Watts TDP, while Intel's specified stock cooler BXT15A (PCG 2015D) is 130 Watts TDP, which is sold separately:
Intel Stock Coolers -
http://www.anandtech.com/show/10500/stock-cooler-roundup-intel-amd-vs-evo-212/3
I would not recommend using a 73 Watt cooler on a 91 Watt CPU. Although it will work, you won't be able to run anything more than just very light workloads, without constantly overheating the 6700K to the point of thermal throttling at 100°C ... very not good. Of course, overclocking will absolutely not be possible.
Somehow, scrape together enough funds to purchase at least a basic cooler such as the Cooler Master Hyper 212 EVO.
CT