* Ireland downgrade pushes euro down toward $1.38
* Dollar index hits highest since May 20
* 2-year U.S. note yields jump to highest in 7 months
* Focus on U.S. Treasury auctions this week (Adds comments, details. Changes byline)
By Vivianne Rodrigues
NEW YORK, June 8 (Reuters) - The euro fell broadly on Monday, hurt by a rating agency's second downgrade on Ireland's sovereign debt, while the dollar extended gains as Treasury yields continued to rise following last week's stronger-than-expected U.S. jobs data.
Ratings agency Standard & Poor's cut Ireland's sovereign credit rating on Monday to AA, its second downgrade in three months. That provided a fresh catalyst to sell the euro, which earlier fell to $1.3806, almost a two-week low.
It also helped the dollar, which rallied since Friday after data showed the United States lost fewer jobs than expected in May. That triggered speculation the Federal Reserve may raise interest rates next year, pushing two-year government note yields to a seven-month high.
"The immediate reaction (to the downgrade) was to take the euro lower, which in any event was trading heavily," said Marc Chandler, senior currency strategist at Brown Brothers Harriman in New York.
The euro fell as low as $1.3806, according to Reuters data, and in afternoon trading was last down 0.4 percent for the day at $1.3910. Against the yen, the euro was down 0.8 percent at 136.79 yen.
"The euro was under pressure after last week's dollar bounce and the downgrade puts further pressure on it now," said Geraldine Concagh, economist at AIB Group Treasury in Dublin.
Technical strategists at Citigroup said fairly strong euro support lies in the $1.3722-$1.3793 area, "which should hold at least on the first attempt."
An index that gauges the dollar's value against six major currencies hit its highest level since May 20. It was last up 0.2 percent at 80.817 after having its best performance since December on Friday.
The dollar slipped 0.3 percent against the yen to 98.33 yen, still close to Friday's one-month high of 98.90 yen on trading platform EBS.
Sterling was last up 0.5 percent at $1.6068 in volatile trading. Earlier, it fell as low as $1.5803 after support for Prime Minister Gordon Brown's ruling Labour Party in European elections plunged to its lowest level in a century, adding to uncertainty about his political future.
HIGHER US YIELDS
Even before Ireland's downgrade, the dollar was buoyed by investors' focus on improving prospects for the U.S. economy, reversing a trend in which the dollar had been sold for higher- risk currencies.
The pace of U.S. job losses slowed sharply last month, the strongest sign to date that the recession is diminishing, even as the unemployment rate hit its highest in nearly 26 years.
The spike in two-year note yields forced more investors to unwind bets against the U.S. currency as some analysts said the dollar's earlier slide had been excessive. Two- and 10-year U.S. Treasury yields hit their highest levels since November in Asia on Monday as Treasuries extended losses.
"Another reason that has intensified the rally in the dollar has been the upward explosion of 10-year yields," said Jessica Hoversen, a fixed income and foreign exchange strategist at MF Global Research, in Chicago. "The spike in Treasury yields improved the dollar's interest rate differential and thus intensified the dollar's gains."
Chuck Butler, president of Everbank World Markets in St. Louis, added that the rise in bond yields has some investors looking ahead, "thinking the Fed is probably going to be finished with its near-zero interest-rate policy by next year."
That view was reinforced on Friday when Atlanta Fed President Dennis Lockhart said the Fed needs to be "anticipatory" and not wait too long before tightening policy.
With currency movements closely following U.S. bond yields, markets will focus on a slew of Treasuries auctions this week totaling $65 billion in debt. (Additional reporting by Steven C. Johnson in New York; Editing by Dan Grebler)