* Front month remains well below recent 21-month high * Weather forecasts mixed for early June * Nuclear power plant outages back above normal By Eileen Houlihan NEW YORK, June 3 (Reuters) - U.S. natural gas futures prices seesawed on either side of unchanged territory early Monday, with the front month contract hovering near the key $4 per million British thermal unit level. With weather forecasts mixed for the next two weeks and nuclear power plant outages only slightly above normal, most traders said a bias remained to the downside after five straight losses. As of 9:25 a.m. EDT (1325 GMT), front-month July natural gas futures on the New York Mercantile Exchange were at $3.987 per million British thermal units, up 0.3 cent, after trading between $3.951 and $4.03. The nearby contract fell nearly 6 percent in last week's holiday-shortened week and was down 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 Sunday called for below-normal temperatures in the eastern two-thirds of the nation and above-normal readings in the western third. Nuclear plant outages totaled 12,900 megawatts, or 13 percent of U.S. capacity, up from 12,400 MW out on Friday and a five-year average outage rate of 12,200 MW, but down from 18,700 MW out a year ago. Baker Hughes data on Friday showed the gas-directed rig count was unchanged for a second straight week at 354. The count posted an 18-year low of 350 three weeks ago. 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 this week's EIA gas storage report range from 78 bcf to 95 bcf versus a year-ago gain of 63 bcf and a five-year average build of 92 bcf for that week.