Yes and no.
First of all: This is not directly related to any game, so you're better off at a Linux forum for help. I recommend you the most popular Linux subreddits on reddit.
1. No ...
... it's not possible in a very convenient way. And to say this at the beginning, you're really better off having a native Steam and a WINE Steam and switch between them when you need either of it.
2. Yes, you can!
Ok, there is a way to do it like you want it. There's a GitHub project called SteamBridge which lets you start WINE Steam games per native Steam. AFAIK you can also use Steam overlay from the native one.
The downside is that as it uses a custom Steam DLL, this clearly breaks the Steam TOS. So especially if you do competitive online-gaming do not use this as this may lead to a VAC ban.
Some users report that they use this a couple of time and never got problems with it, other state they got banned because of it.
You may try it, but I warned you.