To purchase from auction you need to be a US person, which allows you to place a competitive or non-competitive bid at TreasuryDirect.
Brokers handling U.S. Treasuries typically can handle initial auction orders too. That part doesn’t change. TreasuryDirect is just another option for those who are eligible to use it.
As background, TreasuryDirect has no analog in Singapore. For U.S. persons (and possibly a few others) the U.S. Department of the Treasury (through its Bureau of the Public Debt, I believe) operates its own channel for direct public participation in auctions of U.S. Treasuries. U.S. persons are not obliged to use this channel (TreasuryDirect) specifically, but it’s a popular option. In Singapore we have what are called “primary dealers,” and the retail public places their orders through their channels. The U.S. Treasury also offers direct custody services for these bonds/notes/bills/securities. Here in Singapore we don’t have that either. The custody is via CDP or (for SRS and CPF) primary dealers. U.S. Treasuries still have primary dealers, of course.
The T-Bill you are referring to is probably 912796ZN2 28/9-28/12.
I'm looking at the price on IBKR right now, Offer/Bid yield is 5.376%-5.481%. Commission is US$20 (0.2 bp) for US$1M
Each individual U.S. Treasury issue has what’s called a “CUSIP,” a unique identifier similar to an ISIN. 912796ZN2 is an example of a CUSIP.
Chicago-based Firstrade Securities also offers brokerage services for U.S. Treasuries to residents of Singapore. They operate on what’s called a net yield basis, meaning that the price they quote you on a U.S. Treasury is the price you get, commission inclusive.
As an aside, I’d love to know why a resident of Singapore is holding US$1M+ and wanting to park that pile of U.S. dollars in an uninsured fixed deposit, uninsured structured deposit, or (better at least) a U.S. T-Bill. What’s the objective? I’m not aware of any 4 year undergraduate education at any U.S. university (for example) that costs anywhere near US$1M even at list price, even with rather lavish student housing assumptions.