Hi everyone! I really need an advice from all of you. I've found a possibility for a PhD but it's not funded so, together with the professor who offered me the position, I'm writing a project for a scholarship... But it's a very competitive one and I don't know if it will have a positive result!! Do you think that, at the same time, I should search for other PhD?

Yep.

definitely

Thank you TreeofLife and newlease36!! But if I apply to another PhD and I get the position more interesting/paid/whatever, what should I say to this professor? I'm worried about his reaction on losing his time and waste of energy... I don't know if he knows that a student search for a lot of opportunities in order to get a PhD.. I need a book of good manners for PhD applicants!!

Personally I would tell him of my plans. Tell him that your project with him is your first priority and choice (if it is), but that you also will be applying elsewhere to increase your chances of getting funding. He should understand.

Personally I wouldn't tell him at all as I can see no advantage to you in doing so.
If you tell him, you risk him pulling the plug on your favoured application and handing it to someone else.
In theory he should not have a problem with you looking elsewhere but unfortunately the real world is full of people who simply will not be reasonable. You need to ask yourself whether you wish to risk this PhD position by telling him before you know you definitely have funding elsewhere. It is none of his business where you look in the meantime.
Consider what you would do if this was a permanent job. Would you consider voluntarily telling each company who interviewed you that you were considering other jobs before they asked?

This feels familiar... I am sure I once advised this before and then pm133 came on and advised otherwise! I agree that there is no need to tell. In my situation, I did it because I had known the potential supervisor for a couple of years and I wanted to tell her rather than she hear it on the vine and be surprised or whatever (academia is a very small world). It didn't risk my PhD position but I can see how it might be something you'd want to consider.