Is it possible to connect an Xbox One wireless controller to the Xbox 360 wireless receiver for PC?

As others have stated, you cannot connect an Xbox One controller through the Xbox 360 wireless receiver. You can buy the Microsoft Xbox One Controller + Wireless Adapter for Windows 10 ($62.99 USD) that includes the adapter you will need. A better option however would be the bluetooth model as it does not require the rather bulky wireless dongle. This particular model was introduced with the Xbox One S. Pay close attention though, only these specific models of controllers have the universal internal bluetooth adapter. These controllers can be identified by the front face plate which is one solid piece of plastic as seen on the left in this figure:

If you don't have Windows 10, or are experiencing connection issues, refer to my post here where I posted a work-around for such problems. You can purchase the Xbox One Wireless Controller here for $59.99 USD.

Note: that Microsoft no longer sells the model that includes the dongle from their marketplace.

As of now, no. As for the future, very unlikely. Both controllers use a different protocol to communicate with their respective receivers. While it might theoretically be possible, it would need to be done on the receiver side.

Try getting a usb cable to connect the Xbox controller (even though it’s wireless it still has a usb entrance) to the 360 console. This should sync the controller to the Xbox 360 console.

Now you can. New drivers are out and about so download your drivers for PC and pair your XBox One controller. Good luck