TREASURIES-Bonds fall under heavy weight of 10-year supply

* Prices down after less-than-stellar 10-year auction

* 30-year Treasury auction set for Thursday

* 10-year yield highest since October 2008 (adds quote, details of auction)

NEW YORK, June 10 (Reuters) - U.S. Treasury prices fell on Wednesday, sending benchmark yields to eight-month highs, after an auction of 10-year notes heightened concerns about the cost of financing the burgeoning U.S. budget deficit.

It was the first test of the government's long-term borrowing ability since investors began to wonder last month whether the United States' prized AAA credit rating may be living on borrowed time.

By some measures the auction went well, with high demand overall and a proxy for foreign interest, the indirect bidding, very robust, especially for the reopening of a previously issued security.

The main downside was that the high yield at the auction was above market expectations. This "tail," as it is known in the market, showed investors wanted the government to pay a premium to get the bonds sold and tipped the balance for a negative interpretation of the sale.

"The auction would have to be considered on the weak side," said John Spinello, chief fixed-income technical strategist at Jefferies & Co in New York.

The existing 10-year Treasury note US10YT=RR was down as much as a full point in the immediate wake of the auction. It was last trading down 26/32, yielding 3.97 percent, versus 3.87 percent at Tuesday's close.

Before the auction it was down 19/32, yielding 3.94 percent. During the selloff, 10-year yields rose as far as 3.99 percent, their highest since October 2008.

(Reporting by Burton Frierson)