If you are currently an undergraduate, and you want to do a PhD in the US, then you will pretty much need to have research experience before you apply. So how you deal with the pay situation should depend on how much you need this particular research experience.
If you are likely to have other chances to get research experience before you apply for PhDs, or you have already got some experience then maybe you should tell him you would need funding - you could say for example that you are really keen, but you would not be able to afford to cover your costs for a whole summer.
HOWEVER, if you are not likely to have other research experience before you apply, then you should jump at this opportunity to get some. You could tell him that the experience is important for you - so you would take out a loan if he cant fund you, but you would obviously prefer not to have to increase your pile of debt. Hopefully he would pay you anyway in that case.
The research opportunities available to you will depend on your situation. For example if you are in Europe, there is a good chance that it will be very hard to get anyone to pay you to do research before you graduate. Many european students are put off by this, and dont have any research experience before they graduate, and then apply for US PhDs with little success. If you are in the US, chances are your school will have some scheme set up to help you get research experience, and you should therefore have other opportunities unless you are already in your penultimate year.