* Prices fall before $29 bln 7-yr note sale * Bonds pare earlier losses as jobless claims rise * Weakness in inflation-linked debt seen problematic for Fed * Fed buys $1.53 bln of bonds due 2036-2043 By Karen Brettell NEW YORK, May 30 (Reuters) - U.S. Treasuries dipped on Thursday before a new sale of seven-year notes, but pared most of their earlier losses after data showed the number of Americans filing new claims for unemployment benefits unexpectedly rose last week. Investors are increasingly sensitive to economic data since Federal Reserve Chairman Ben Bernanke said last week the U.S. central bank may decide to taper its program of buying Treasuries and mortgage-backed securities within the next few Fed policy meetings if data shows the economy is gaining steam. Yields have surged this month on improving economic sentiment and as investors sold bonds on concerns about losses should the Fed pull back on its massive bond purchases. Data also showed that a drop in government spending dragged more on the U.S. economy than initially thought in the first three months of the year. "We're very hyper-sensitive to the employment side of things, though the GDP is an old number," said Tom Tucci, head of Treasuries trading at CIBC in New York. The recent backup in rates may help the Treasury sell $29 billion in seven-year notes on Thursday, the final sale of $99 billion in new coupon-bearing debt this week. The government saw strong demand on Wednesday for a $35 billion auction of five-year notes. Some of that demand may ebb, however, if investors seeking intermediate-dated debt bought yesterday's five-year notes at the expense of today's seven-year offering. The dramatic surge in Treasuries yields this month is leaving many investors struggling to know whether the rise will continue, or if the market is oversold. Data next week that will show how many jobs employers added in May is seen as the catalyst for deciding the next large rate move. "We've reached an uneasy equilibrium in the market, where we can easily see the prospect of much lower or much higher rates," said Aaron Kohli, an interest rate strategist at BNP Paribas in New York. Benchmark 10-year notes were last down 3/32 in price to yield 2.13 percent, after rising as high as 2.16 percent before the data. The yields have jumped from 1.61 percent at the beginning of May, and reached a 13-month high of 2.24 percent in overnight trading on Wednesday. A dramatic selloff in the inflation-linked bond market, however, may complicate the Fed's objective of encouraging borrowing in order to fund economic expansion. The so-called real yield, which is the amount a bond yields after accounting for inflation, has been trading at negative levels for the past few years, which has made it attractive for companies and other borrowers to issue debt at very cheap levels. "This is the Fed's lever for making the economy go," said Kohli. After the recent selloff, however, "the market seems to be placing no possibility at all on the prospect of inflation." Treasury Inflation-Protected Securities (TIPS) staged a dramatic selloff on Wednesday even as Treasuries prices rose. Ten-year real yields are now trading at around negative 15 basis points, meaning a borrower would earn 15 basis points per year from issuing debt, in from around negative 72 basis points at the beginning of May. The release of April's Personal Consumption Expenditures index on Friday, the Fed's favored inflation gauge, will be closely watched for a further drop in price inflation. The index has fallen to a 3-1/2-year low of 1.0 percent. The Fed bought $1.53 billion in debt due 2036 and 2043 on Thursday as part of its bond purchase program.