* Focus on testimony of Fed's Yellen
* News on Trump Jr's e-mails briefly pushes U.S. yields lower
* Fed's Brainard wants to move cautiously on interest rates
* U.S. three-year note auction post mixed results
(Adds comment, 3-year note auction results, updates prices) NEW YORK, July 11 (Reuters) - U.S. Treasury yields slipped in choppy trading on Tuesday after Federal Reserve officials, on the eve of Fed Chair Janet Yellen's congressional testimony on monetary policy, expressed doubts about further interest rate hikes due to low inflation. Fed Board Governor Lael Brainard said she believed the U.S. central bank should begin unwinding its balance sheet soon, but she would want to "move cautiously on further increases in the federal funds rate" to help push inflation higher toward its target. Philadelphia Fed President Patrick Harker, a voter on the Federal Open Market Committee, also echoed Brainard's sentiment. In an interview with the Wall Street Journal on Tuesday, Harker said if inflation did not move toward the Fed's 2 percent target, then that would be a reason to hold off raising rates. Jim Vogel, interest rates strategist, at FTN Financial in Memphis, Tennessee, said those comments sounded like both officials were leaning toward a pause in the Fed's tightening, after raising rates twice so far this year. He added that chances of a December rate hike were now down to 50-50 percent again, after hitting 60-40 percent last week in favor of a rate hike. Long-dated yields also dipped after President Donald Trump's eldest son released an email chain, which referred to a top Russian government prosecutor as offering the Trump campaign damaging information about Democratic rival Hillary Clinton.
"This suggests that allegations the Trump campaign colluded with Russia are not yet settled," said Gennadiy Goldberg, interest rates strategist at TD Securities in New York. "At the margin, this draws attention away from President Trump's economic agenda." The market's focus, however, remained on Yellen, who will deliver her semi-annual testimony before the House Financial Services Committee on Wednesday and the Senate Banking Committee on Thursday. Investors will be looking to Yellen for clues on when the Fed will start reducing its massive balance sheet. In late trading, the benchmark 10-year Treasury note yielded
2.364 percent , down slightly from 2.371 percent late
on Monday. Yields on U.S. Treasuries with maturities from 2 to 10 years were also modestly lower.
U.S. 30-year yields were at 2.924 percent,
compared with Monday's 2.923 percent. The Treasury's auction of $24 billion in three-year notes, meanwhile, was mixed. The note fetched a high yield of 1.573 percent, just below the 1.576 percent rate at the bid deadline. Bids totaled nearly $68.9 billion for a 2.87 bid-to-cover ratio, a little below last month's 3.00, but slightly higher than the 2.80 average.
(Reporting by Gertrude Chavez-Dreyfuss; Editing by G Crosse and Andrew Hay)