* Front month down 4 percent last week, second weekly loss * Weather outlooks turn warmer for mid- to late-June * Nuclear power plant outages well below normal By Eileen Houlihan NEW YORK, June 10 (Reuters) - U.S. natural gas futures seesawed on either side of unchanged early on Monday, as the return of weekday industrial demand and warm weather outlooks for late-month mixed with near-term mild conditions in the Northeast, a quiet tropical front and below-normal nuclear plant outages. As of 9:26 a.m. EDT (1326 GMT), front-month July natural gas futures on the New York Mercantile Exchange were at $3.836 per million British thermal units, up 0.8 cent, after trading between $3.815 and $3.864. The contract hit a nearly three-month low of $3.814 on Friday after climbing to a 21-month high of $4.444 in early May. The latest National Weather Service six- to 10-day forecast and the eight- to 14-day outlook, both issued on Sunday, called for above-normal temperatures for most of the nation, with below-normal readings only along the West Coast and in New England. Nuclear plant outages totaled just 6,600 megawatts, or 7 percent of U.S. capacity, down from 11,200 MW out on Friday, 11,700 MW out a year ago and a five-year average outage rate of 9,500 MW. The U.S. National Hurricane Center said tropical cyclone formation was not expected during the next 48 hours. Early injection estimates for Thursday's gas storage report from the U.S. Energy Information Administration range from 83 bcf to 107 bcf versus a 66-bcf build during the same week last year and a five-year average rise for that week of 84 bcf. Baker Hughes data on Friday showed the gas drilling rig count held steady for a fourth straight week at 354, just above the 18-year low of 350 hit in early May.