Could this cause the tyre to blow?
The answer is yes.
When springs break, the larger piece often remains "in place" and under compression by the vehicle's weight - the issue caused by this is that the car will ride lower on that corner. In your case, the car is riding low enough for the tire to be contacting the inside of the wheel well liner. This can be an immediate problem - as you drive the vehicle, the suspension will move, but as it's moving, it'll be forcing the (turning) tire against the inside of the fender. This can easily cause a tire to rupture.
Further, even though the larger chunk of remaining spring will often settle under the car's weight, the smaller, broken-off piece can have a mind of it's own. I've seen cases where the smaller piece is ejected from the vehicle under high energy - landing 30 yards away! Even if the smaller piece isn't totally ejected from the vehicle, it may shift or bounce around as you drive, and it can easily rip open the inner sidewall of the tire, or become jammed in the suspension or steering, causing loss of control.
Even when your car is sitting still, suspension spring store a tremendous amount of energy as they're compressed by the vehicle's weight. It's not something you want to fool around with.