* Markets bet bill on $2,000 aid check will not pass in Senate * U.S. yield curve modestly steeper * U.S. 7-year note auction shows lackluster results * German, Italian 10-year yields flat on day in thin trade (Repeats to additional subscribers without any changes to text) NEW YORK, Dec 29 (Reuters) - U.S. Treasury yields were flat to modestly higher on Tuesday in thin trading as U.S. stocks surrendered gains, with investors betting a heftier federal stimulus check was unlikely to get the nod in the Republican-led U.S. Senate. Legislation on a $2,000 stimulus check is in the Senate’s hands on Tuesday after President Donald Trump signed a $2.3 trillion pandemic aid and spending package on Sunday. Risk appetite ebbed during the latest hour, with yields turning lower for some maturities, as investors digested comments from congressional leaders on the hotly debated stimulus. U.S. Senate Majority Leader Mitch McConnell on Tuesday blocked immediate consideration of a measure to increase COVID-19 relief payments to $2,000, but he suggested the Senate would at least examine the issue. “The wall among Senate Republicans to stick to $600 impact payments is cracking, but the majority leader could have the cement to hold it together,” said Jim Vogel, senior rates strategist at FHN Financial in Memphis, Tennessee, in his latest research note. “Time is running out, maybe,” he added. As this developed, the Treasury’s U.S. 7-year note auction, the last coupon supply for 2020, showed tepid results, weighed down by the increase in auction size, and in line with the underwhelming outcomes for the 2-year and 5-year offerings on Monday. The high yield was at 0.662%, slightly higher than the “when-issued” or expected level of around 0.660% at the bid deadline, which meant investors demanded a little more yield to own the paper. There were $136.6 billion in bids for a 2.31 bid-to-cover ratio, a gauge of demand. That ratio was lower than last month’s 2.37 and the 2.47 average, according to analysts. Indirect bidders, which include foreign central banks, took 60.3% of supply, lower than November’s 65.4% uptake, and the 63.1% average. The U.S. yield curve was slightly steeper on the day, with the overall steepening trend still intact amid increased risk appetite trending the last few sessions, as well as expectations of higher inflation. The yield spread between 2-year and 10-year notes was last at 80.60 basis points. Last week, that curve hit its widest in more than three years. Since late July, the gap between U.S. 2-year and 10-year yields has widened by roughly 50 basis points. In afternoon trading, the U.S. benchmark 10-year yield was flat at 0.934%, from 0.933% late on Monday. Post-auction, U.S. 7-year note yields were last down at 0.645%, from 0.648% on Monday. U.S. 30-year yields rose to 1.673% from Monday’s 1.669%. On the front end of the curve, U.S. 2-year yields inched lower to 0.128% from 0.133% on Monday. Breakeven rates on 10-year TIPS, which measure expected annual inflation for the next 10 years, slipped to 1.963%, from Monday’s 1.967%. In the European bond market, benchmark German 10-year government bond yields were flat at -0.573% in light trading, while benchmark Italian 10-year bond yields slipped to 0.521%, from 0.522% on Monday. Volume is likely to remain thin in Europe this week, with German stock markets already closed on Thursday and all markets shut on Friday for New Year’s day. (Reporting by Gertrude Chavez-Dreyfuss; Editing by Richard Chang)