* Front month still well below recent 21-month high * Weather moderating over most of the country * Nuclear power plant outages remain above average * Coming up: Baker Hughes gas drilling rig data Friday By Eileen Houlihan NEW YORK, May 10 (Reuters) - U.S. natural gas futures seesawed on either side of unchanged early on Friday, with a slight bias to the downside as mild, spring-like weather blanketed the country and another larger-than-expected weekly inventory build was reported a day earlier. Still, prices remain well under last week's 21-month high. A long, cold winter put a huge dent in inventories and lingering cool weather this spring led to a slow start to the injection season. But the onset of milder spring weather starting this week has curbed late-season heating demand before heavy cooling loads kick in. As of 9:24 a.m. EDT (1324 GMT), front-month June natural gas futures on the New York Mercantile Exchange were at $3.969 per million British thermal units, down 1.4 cents, after trading between $3.937 and $4.01. The contract hit a one-month low of $3.883 on Thursday, after climbing to a 21-month high of $4.444 last week. The National Weather Service's latest six- to 10-day forecast issued on Thursday called for above-normal temperatures for about the western half of the nation and below-normal readings in the Southeast. Near-normal temperatures were expected for the Midwest, mid-Atlantic and Northeast. Nuclear plant outages totaled 19,700 megawatts, or 20 percent of U.S. capacity, down from 20,900 MW out on Thursday and 21,300 MW out a year ago, but up from a five-year average outage rate of 19,600 MW. ANOTHER LARGE WEEKLY BUILD, BUT STOCKS BELOW NORMAL Thursday's gas storage report from the U.S. Energy Information Administration showed domestic inventories rose last week by 88 billion cubic feet, above Reuters poll estimates for an 83-bcf build, a year-ago gain of 30 bcf, and a five-year average build of 69 bcf for that week. But inventories, which started the injection season about three weeks later than expected due to the cold spring, are at 1.865 trillion cubic feet, more than 28 percent below last year and 5 percent below the five-year average. Early injection estimates for next week's EIA report range from 87 bcf to 108 bcf versus a 56-bcf build in the same week last year and a five-year average rise for that week of 83 bcf. Traders awaited the next Baker Hughes gas drilling rig report, due later on Friday. Data last week showed the number of rigs drilling for natural gas in the United States fell by 12 to an 18-year low of 353.