* Milder U.S. weather next week expected to slow demand * Nuclear outages still running above normal, lend support * Big inventory draw expected Thursday also limits downside * Coming up: Reuters natural gas storage poll Wednesday By Joe Silha NEW YORK, April 2 (Reuters) - Front-month U.S. natural gas futures ended lower on Tuesday for a third straight session, pressured by prospects that milder weather next week, particularly in the East and South, will finally bring an end to heating-season demand. Cold late-winter weather and above-average nuclear plant outages helped trim record high inventories at the start of the heating season to near normal levels and drive futures prices up nearly 30 percent since mid-February. But despite chilly temperatures this week, many traders do not see much upside in prices, with milder spring weather on the way and production still flowing near a record peak. "We're seeing moderation in the weather, and there's been some trepidation at the $4 (per mmBtu) level, but I think there's still support at about $3.93, and people are expecting a big inventory draw on Thursday," said Matt Smith, commodity analyst at Schneider Electric in Kentucky. Front-month gas futures on the New York Mercantile Exchange ended down 4.6 cents, or 1.1 percent, at $3.969 per million British thermal units after trading between $3.943 and $4.044. The front contract, which hit a 19-month high of $4.121 late last week, has lost 2.4 percent in the last three sessions. Nearby futures, which have posted gains for six straight weeks, finished March about 15 percent higher and ended the first quarter up about 20 percent. Chart watchers noted the near steady climb in open interest has backed much of the recent price rise, indicating that new buying, not short covering, has been fueling the upside. Futures-only open interest on Monday hit a record high for the eleventh straight session, climbing to 1,455,630 contracts. While the Plains and upper Midwest could remain cool next week, forecaster MDA Weather Services expects above to much above normal temperatures to stretch from the lower Midwest to the East and South. ANOTHER STRONG STORAGE DRAW EXPECTED U.S. Energy Information Administration data last week showed total domestic gas inventories fell to 1.781 trillion cubic feet, just 61 billion cubic feet, or 3.5 percent, above the five-year average for that week. Most traders expect storage to drop below the five-year average in Thursday's EIA report, which would be the first time below that benchmark since September 2011. Withdrawal estimates range from 65 to 108 bcf, with most in the high-80s. Inventories rose 43 bcf during the same week last year. The five-year average for that week is a 4 bcf build. Stocks began winter at a record 3.929 tcf, but about 2.15 tcf of gas has been pulled from storage so far this heating season, or 45 percent more than last year at this time. OUTPUT STARTS TO SLOW? EIA data on Friday showed that gross natural gas production in January fell for the second straight month. Output also dropped below year-ago levels for the first time since February 2010, but it's still unclear if recent monthly declines were due to well freeze-offs from the cold or producers curbing dry gas flows because prices were not that attractive. Baker Hughes data Thursday showed the gas-directed rig count fell last week for the fourth time in five weeks, dropping by 29 to 389, its lowest since May 1999. EIA last month was still expecting marketed gas output in 2013 to hit a record high for the third straight year.