I have saved about US$40,000 in cash and investments in order to pay for my LLM.
I thought my chances of acceptance would improve if I could show that I had enough funds to cover tuition and minimal expenses. However, will this DISadvantage me in terms of receiving some form of financial aid?
How much to disclose?

It depends. Some law schools only award need-based scholarships. Yale, for instance. Others may also award merit-based ones. I have scholarships worth 60, 000 U.S.$ that have been awarded in Finland. For that reason, Yale only gave me 2500 U.S.$. Cornell, on the hand, would have awarded me with a full ride scholarship (worth 42, 500 U.S.$). Your own fund might therefore risk your chances of getting financial aid. If you choose to disclose only part of your funding, you obviously take a risk. And remember, having funding might increase your chances of making it to the law school you wish to attend (though admissions are supposedly need-blind).

Thanks tmalmine! and congragulations on your scholarships and admissions!
Do you by any chance know how Harvard or Columbia would respond? ie whether their admissions process is "need-blind" and whether their financial aid is merit or needs based?
Thanks again for your time.

Harvard is pretty generous with their financial aid (or so I've heard), and might give you something despite your own funding (if they want you, that is). I was accepted to Columbia, but not given any financial aid, that's all I can say. Obviously every law school says its admissions are need-blind, but I don't believe it. I made it to every law school I applied, and I think it might have something to do with my finances. But this is just my hunch. If I were you, I would be honest with my financial situation, although it might cost you some money. Good luck, whatever you decide to do!