* Front month remains well below recent 21-month high * Weather forecasts mixed for early to mid-June * Nuclear power plant outages remain above normal * Coming Up: EIA natgas storage data on Thursday By Eileen Houlihan NEW YORK, June 4 (Reuters) - U.S. natural gas futures seesawed on either side of unchanged territory early Tuesday, with the front month contract again hovering near $4 per million British thermal units, a key psychological level. With weather forecasts mixed for the next two weeks and nuclear power plant outages only slightly above normal, most traders expect the bias to remain to the downside. As of 9:24 a.m. EDT (1324 GMT), front-month July natural gas futures on the New York Mercantile Exchange were at $4.003 per million British thermal units, up 1.2 cents, after trading between $3.965 and $4.033. The nearby contract edged up less than 1 cent on Monday, after falling nearly 6 percent in last week's holiday-shortened week and more than 8 percent in May. It hit a one-month low of $3.883 on May 9 after climbing to a 21-month high of $4.444 on May 1. The latest National Weather Service six- to 10-day forecast issued on Monday again called for below-normal temperatures for about the eastern half of the nation and above-normal readings in most of the western half. Nuclear plant outages totaled 12,800 megawatts, or 13 percent of U.S. capacity, down from 12,900 MW out on Monday and 17,500 MW out a year ago, but up from a five-year average outage rate of 11,900 MW. Data last week from the U.S. Energy Information Administration showed inventories rose the prior week by 88 billion cubic feet to 2.141 trillion cubic feet. Stocks are nearly 24 percent below year-ago levels and nearly 4 percent below the five-year average. Early estimates for Thursday's EIA gas storage report range from 80 bcf to 96 bcf versus a year-ago gain of 63 bcf and a five-year average build of 92 bcf for that week.