I have a feeling it won't, but it would be nice to not have to buy and fit a bike carrier!

It's the 3 door version if that matters, the seats don't fold all the way down.

Considering I used to get two (road/hybrid) bikes in my 500 easily enough, I'd be amazed if you couldn't in the punto.

The seats do fold down you lifted the seat base?

The seats fold nearly flush, have to lift the section where your bum sits... Lol and then the back section will go further down ?...

Managed to fit 2 BMX's in my 3 door with the seats fully down (Pull the back of the bum part of the seat towards you so it rests against the front seats and pull the back section to where the bum part just was) with a Subwoofer still in the boot, so i'm sure a single bike is more than possible

I've had 3 in my 3 door gp with the seats down and front wheels off.

Yes a bike will fit in a grande punto. If your rear seats have head rests remove these then it will fold all the way done.
And it's easier if you remove your wheels from the bike.

Hope this helps.

thanks guys, didn't know the seats folded all the way down like that

In the past I've de wheeled the bike wrapped a tarp between the back seats and up the back of the front seats and put the bike in the rear foot well with the wheels there too. Just make sure the chain ring or peddles and rear mech doesn't rub

My mountain bike fits with both wheels removed. Might fit with the rear still on but I've not tried.

P.s. No need to remove the rear headrests when folding the seats. The rear seat bases have a cut out on the underside into which the headrests fit. You do need to tip the front seats forward during folding though to get the headrests into the cut out.