* Spanish, Italian yields lower but Monday's rally falters * German Bunds supported by U.S. budget talks gridlock * Peripheral rally may yet have further to run - KBC By William James and Emelia Sithole-Matarise LONDON, Dec 4 (Reuters) - Recent volatility in euro zone government bonds eased on Tuesday, leaving yields on both low-risk and higher-yielding debt slightly lower on the day as U.S. worries offset a positive mood in the bloc. After big gains on Monday the rally in peripheral euro zone debt, fuelled by optimism that Greece will meet the conditions necessary to secure its latest bailout tranche, looked to be running out of steam. At the same time, German debt recouped a large part of Monday's losses as demand for the safety of Bunds remained intact thanks to U.S policymakers' wrangling over how to avoid spending cuts and tax hikes seen likely to spark a recession. "We've got a 'risk-on' tone in Europe but a macro global 'risk-off' move as people worry about what's going on in the U.S., and Bunds benefit a bit from the U.S. backdrop," said Lyn Graham-Taylor, strategist at Rabobank in London. German Bund futures settled 23 ticks higher at 142.74, holding within the 141.84-143.48 range established during November. In the periphery, Italian 10-year yields settled 3 basis points lower at 4.42 percent while the Spanish equivalent was last down 1 bp at 5.26 percent. That was only a slight move compared with recent large gains spurred by the belief that a Greek debt buy-back would be successful thanks to generous terms announced on Monday, and would pave the way for Athens to receive aid funds. In the previous session Italy's borrowing costs fell to fresh two-year lows and its 10-year yield premium over Bunds dropped below 300 bps for the first time since March. "We think that the big items such as an imminent bankruptcy in Greece are off the table. The overall climate is in favour of investors looking for yield everywhere and so the way of least resistance is for a narrowing of yield spreads," Piet Lammens, a strategist at KBC, said. Greek bond prices were slightly lower after Monday's sharp rally, ahead of the results of the bond buyback due at the end of the week. BUND OUTLOOK German 10-year yields were down 1.7 bps at 1.39 percent, well within their recent 1.30-1.70 percent trading range and traders saw little scope for yields to break the top of that range given concern about congressional gridlock in U.S. budget talks. Investors were doubtful that U.S. lawmakers will reach a deal in time to avert a recession-inducing fiscal tightening in the world's biggest economy early next year after the White House dismissed Republican proposals for steep spending cuts. "The fiscal cliff debate rumbles on ... The growth implications if they don't get something sorted out are fairly severe and as long as that remains in play, core markets remain reasonably well supported," a trader said.