Few businesses currently accept Bitcoins as payments, but its supporters hope it will one day be a widely used alternative to the dollar and other government-controlled currencies.
The value of a Bitcoin has fluctuated wildly in recent months, rising from $20 to $266 before suddenly plunging to about $130 in April.
"It seems to me there is a prudential regulation that we may want to insert if people have the possibility of not being backed up, if things aren't cleared, if there's no margin requirement," Chilton said. "You don't want it to be just a house of cards that can impact people."
Andrew Ross Sorkin, a CNBC anchor, pressed Chilton on whether the government is even capable of regulating Bitcoins.
"It seems like one of those instruments that almost can't be regulated because there is no central repository, central bank, central anything," he said.
"If it's traded in U.S. commerce, if it's traded on our computers, we could potentially oversee it," Chilton argued.