It's a problem most people can't relate to: Warren Buffett has too much money. More specifically, Berkshire Hathaway (NYSE:BRK-A) (NYSE:BRK-B) has too much cash. More than $35 billion at the end of last quarter, to be exact. And it's all sitting around earning a rate of return that we can presume is less than inflation.

Consequently, as Buffett put it in 2006, "We continue ... to need 'elephants' in order for us to use Berkshire's flood of incoming cash. Charlie [Munger] and I must therefore ignore the pursuit of mice and focus our acquisition efforts on much bigger game."

Of course, Buffett's idea of an elephant is everyone else's idea of the global population of elephants. Last summer, he told CNBC he's looking to make deals in the $40 billion to $60 billion range.

Shhh ... be vewy, vewy quiet. I'm hunting ewephants ...
Amid the negativity of the last year, I'm willing to bet that Buffett has been snooping around the financial sector for bargains. Just last quarter, Berkshire added to its existing stakes in Wells Fargo (NYSE:WFC) and US Bancorp (NYSE:USB).

But let's take a look at another Buffett favorite that's been clobbered lately: American Express (NYSE:AXP). With a market cap of around $43 billion, it's no less than an elephant. Buffett has treasured it for decades, and best of all, its shares' 36% drop in the past year might make it a Berkshire buyout candidate.

But does he want it?                                                          
Few investors have more AmEx experience than Buffett. Back in the '60s, when he ran a hedge fund, Buffett had the audacity to put 40% of his money in AmEx stock at one time. The idea today isn't much different from what it was back then: Consumers -- particularly the wealthy -- love using AmEx for the status it conveys when they whip the card out.

But unlike Visa (NYSE:V) or MasterCard (NYSE:MA), AmEx owns every part of the transaction, including the financing that's causing investors to flee. But that just gives Buffett an opportunity to swoop in while others panic. Given his 40-year interest with AmEx, it's safe to assume that Buffett wouldn't mind owning the entire company.

Could he buy it?                                                          
Absolutely. In April, American Express passed an amendment making it easier to sell the entire company. The amendment reduced the percentage of shareholder votes needed to approve a sale from 66.6% to 50%. Berkshire already owns around 13% of American Express, so the effective percentage of shareholder votes needed to approve a Berkshire buyout is just 37%.

How much would he pay?                                                
Please don't confuse me for someone who thinks he knows -- I wouldn't try to outsmart Buffett any sooner than I'd try to outrun a cheetah with raw meat strapped to my back. Nonetheless, with a current market cap of $43 billion, it seems rational that Buffett could offer a 20% premium -- enough to bring AmEx's market cap to $50 billion -- and still be happy with the price.

Net income for the past three years at AmEx was $4.0 billion, $3.7 billion, and $3.7 billion, respectively. Let's assume Buffett makes a run at the company for $50 billion, or around $44 per share. Even with that premium, he'd be paying just 13 times the average of the past three years' earnings -- not bad for a company in an industry undergoing booming global growth as consumers switch to a cashless culture. On top of everything else, with Berkshire as its parent, AmEx could eliminate needless costs such as shareholder services and Sarbanes-Oxley regulation.

Can he afford it?                                                        
Yes, Berkshire could easily manage a $50 billion buyout without even issuing a single share of stock. On top of the $35 billion in cash, it has a portfolio of common stocks worth $75 billion (at the end of last year) that could quickly be reduced to raise billions in cash. No problems there.                                                                      

But ... will he do it?                                                       
That's the big question. I'm not making predictions here; I'm just trying to show that AmEx could be on Berkshire's list of targets. To date, there hasn't been a peep of information suggesting a buyout in the works. This is purely speculation here, people.

In all likelihood, Buffett is probably more interested in elephant hunting outside public markets, where the barrage of media attention and rumor-milling is more subdued -- such as we saw in his deal to purchase a huge stake in Marmon earlier this year. What will be interesting to see, however, is how the markets would react if Buffett scooped up a company in the battered financial sector. Until he pulls the trigger, rest assured that plenty of investors will be keeping an eye on this elephant hunt.

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Fool contributor Morgan Housel owns shares in Berkshire Hathaway. US Bancorp is an Income Investor selection. Berkshire and American Express are Inside Value recommendations. Berkshire Hathaway is also a  Stock Advisor pick. The Fool owns shares of Berkshire and American Express and, if you hadn't noticed, has a disclosure policy.