When you go to the Free Games With Gold section of the Xbox360, it also shows the free games for the Xbox One. This is clearly meant to encourage you to buy a Xbox One. And there's nothing wrong with that.

However, if you click on one, it loads a Xbox One interface (sorta). The free game for the Xbox One now says "Get This Game". Clicking it causes a short load icon and the text changes to "You Have This Game."

If I'm understanding this correctly, this means that Xbox 360 users can "buy" free Xbox One Games (of the month) via their 360. Thus allowing them to already own the game once they purchase a Xbox One and attach their account to it.

Xbox.com ALSO gives you this option, regardless of which Xbox console you own.
http://www.xbox.com/en-GB/live/games-with-gold?xr=shellnav

It needs to be noted that I have already attached my account to a Xbox One via my former room-mate's console. So that might have something to do with my account having these options. But if you only have a Xbox 360 (or X1), then it's worth checking out at least. Better to own these games when they are free then when the sale ends. ;)