I may have an unusual perspective on this, since mathematics Ph.D. programs in the U.S. generally fund all their students (at least through teaching). In this context, I'd recommend against even mentioning the possibility of self-funding:

1. It sounds unconfident, like you don't expect to receive funding, and this is not a positive impression to convey.

2. Letting Ph.D. students go into debt feels exploitative, especially in a program in which most or all of the other students are funded, and letting wealthy people buy their way into graduate school is also troubling.

3. There will be worries about how sustainable your self-funding is. In a year or two, your money or willingness to go into debt may run out, and the department would have to choose whether to come up with money for you or kick you out. That's an unpleasant choice, and the admissions committee may even worry that you are deliberately aiming for this situation (in the hope that they won't be willing to kick you out once you've joined the department).

In the sort of programs I'm familiar with, announcing in advance that you would be willing to self-fund a Ph.D. won't increase your chances of admission. I've only seen this come up very rarely, and the admissions committee has simply ignored the offer of self-funding and made a decision of acceptance with funding or rejection. I can easily imagine that it could decrease the chances of admission a little, although I haven't seen a clear-cut case of that.

By contrast, if you are awarded a prestigious fellowship with guaranteed funding for a certain number of years, you should certainly make sure admissions committees know about it. That doesn't have any of these disadvantages. However, it's not in your best interests to offer to pay your own way unless that's a common and respected approach in your field.

I may have an unusual perspective on this, since many mathematics Ph.D. programs in the U.S. fund all their students (at least through teaching). In this context, I'd recommend against even mentioning the possibility of self-funding:

1. It sounds unconfident, like you don't expect to receive funding, and this is not a positive impression to convey. You don't want to give the impression that you feel your own application is subpar, and that you're bargaining to try to turn a rejection into an unfunded acceptance. This might be a reasonable approach at the end of the process, if you're stuck on a waiting list, but it's a problematic way to begin.

2. Letting Ph.D. students go into debt feels exploitative, especially in a program in which most or all of the other students are funded, and letting wealthy people buy their way into graduate school is also troubling.

3. There will be worries about how sustainable your self-funding is. In a year or two, your money or willingness to go into debt may run out, and the department would have to choose whether to come up with money for you or kick you out. That's an unpleasant choice, and the admissions committee may even worry that you are deliberately aiming for this situation (in the hope that they won't be willing to kick you out once you've joined the department).

In the sort of programs I'm familiar with, announcing in advance that you would be willing to self-fund a Ph.D. won't increase your chances of admission. I've only seen this come up very rarely, and the admissions committee has simply ignored the offer of self-funding and made a decision of acceptance with funding or rejection. I can easily imagine that it could decrease the chances of admission a little, although I haven't seen a clear-cut case of that.

By contrast, if you are awarded a prestigious fellowship with guaranteed funding for a certain number of years, you should certainly make sure admissions committees know about it. That doesn't have any of these disadvantages. However, it's not in your best interests to offer to pay your own way unless that's a common and respected approach in your field.