Treasury's Bureau of Fiscal Management said it would auction $119 billion of 3-month and 6-month bills on Tuesday, with the sales formally settling two days later on June 1. It will also sell a $50 billion cash management bill, maturing in November, that settles the same day.

That's the date that Treasury Secretary Janet Yellen has repeatedly pointed to as a possible deadline for the government running short of the funds to cover all of its obligations unless a deal between the White House and congressional Republicans is reached to lift the $31.4 billion U.S. debt cap.

According to research firm Wrightson ICAP, the bureau would not typically announce an upcoming sale "until it is certain that it has room under the debt ceiling for the new securities."

Gennadiy Goldberg, senior rates strategist at TD Securities in New York, said the fact that Treasury proceeded with the announcements on Thursday "does suggest that the Treasury probably has cash to settle the security. They have suggested in the past that they would not announce auctions that they did not believe they had the means to settle. So I do think that's a positive note."

"But," Goldberg continued, "that's really where the positivity stops because really we know it's very likely in the first two weeks of June the Treasury will run out of cash."

Treasury is already "scraping the bottom of the barrel" in terms of its cash balance, he said.

Despite Yellen's remarks that the X-date could be as early as June 1, most Wall Street firms estimate the date that Treasury will finally run out of room under the extraordinary measures it has employed for months to stretch its borrowing capacity is more likely between June 6 and June 9.

As of Wednesday, the government's cash balance was $49.47 billion, Treasury said on Thursday afternoon, down from $76.55 billion on Tuesday, and from $68.33 billion a week earlier.

But June 1 is the date that some $117 billion of T-bills matures. With so much uncertainty in the market about the X-date, the yield on that issue had rocketed to a record high above 7% on Wednesday. It tumbled on Thursday after the Treasury auction announcement for the bills set to settle June 1 and as debt ceiling negotiations showed some signs of progress, and last yielded 5.79%.

With the matter growing more urgent by the day, talks between the Biden administration and Republicans in the U.S. House of Representatives have shown progress and the outline of a deal may be taking shape, Reuters reported on Thursday. The two sides are just $70 billion apart on a total figure that would be well over $1 trillion, according to a source.