TREASURIES-Bonds gain with safe-haven support

* Bonds' safe-haven appeal bolstered by stock futures

* Decent demand seen in Tuesday's two-year note auction

* Consumer confidence data also set for Tuesday morning

NEW YORK, July 28 (Reuters) - U.S. Treasury debt prices rose on Tuesday as stock futures pointed to a lower open on Wall Street and bolstered the safe-haven appeal of government debt.

Gains were limited however as investors nervously awaited the auction of $42 billion of two-year Treasury notes on Tuesday afternoon as part of a record-large spate of new debt supply this week.

"Stock futures are the leading cause" for higher Treasury debt prices," said William Larkin, fixed income portfolio manager at Cabot Money Management in Salem, Massachusetts, adding, "We ran so fast with stocks that there is probably going to be some profit taking, which is beneficial to bonds."

The benchmark 10-year Treasury note US10YT=RR was trading 5/32 higher in price for a yield of 3.70 percent, down from 3.72 percent late on Monday. Benchmark yields, which move inversely to prices, climbed to 3.77 percent -- the highest in over a month -- on Monday as worries over pending supply chipped at Treasury prices.

But those concerns eased somewhat on Tuesday, with many analysts expecting relatively solid demand for the two-year notes, as their shorter maturity makes them less susceptible to economic uncertainty.

"The two-year is expected to go off without a hitch because it is the optimal part of the curve," Larkin said.

The Treasury will auction $39 billion of five-year notes on Wednesday, and $28 billion of seven-year notes on Thursday. It sold $6 billion of 20-year inflation-protected securities on Monday, bringing this week's total coupon sales to $115 billion.

In addition to Tuesday's auction, investors were waiting for data on July consumer confidence due later in the morning.

Two-year Treasury notes US2YT=RR were trading unchanged in price for a yield of 1.05 percent, while 30-year bonds US30YT=RR were 10/32 higher for a yield of 4.61 percent, down from 4.63 percent late on Monday. (Reporting by Chris Reese; Editing by Kenneth Barry)