Would an i5-6500 Bottleneck an RTX 2070

Nov 25, 2018
I plan on upgrading my PC's current 1080p 60hz monitor to a 1440p 165hz monitor and upgrading my GTX 970 to an RTX 2070. I believe an RTX 2070 should be able to run games at 2k 165hz but I am not sure if my current i5-6500 will bottleneck an RTX 2070. I will be playing games such as Call of Duty: Black Ops 4, Rainbow Six Siege, Fallout 4, etc. I also have an ASUS Z170 Pro Gaming motherboard, so the highest CPU upgrade I could make would be to a core i7-7700k. Would my old CPU bottleneck my new GPU, and if so what CPU (and possibly motherboard) should I upgrade to?

Edit: I should have mentioned that i currently have 2x 8gb DDR4 ram (16gb total)