BEAUMONT, Texas, June 10 (Reuters) - A lockout of union workers at Exxon Mobil Corp's third-largest U.S. refinery continued without resolution after negotiators for the United Steelworkers union and Exxon met on Thursday for the first time in six weeks.

The labor contract covering 650 union workers at Exxon's 369,024 barrel-per-day (bpd) Beaumont, Texas, refinery and lubricants facility expired earlier this year. Exxon locked out the union workers, citing worries over a walkout, and is operating the 2,700-acre (11-sq km) complex with managers and replacement workers.

The two sides discussed their contract terms and agreed to meet again next week, said Darrell Kyle, president of USW Local 13-243 at the refinery and lubricant plant, which produces Mobil 1 motor oil.

"We went through some things to see if they had room to move," said Kyle. "They said they do, but they have no reason to move now."

Exxon said its current offer remains in effect.

"No additional proposals were exchanged in today’s meeting with the union," said spokesperson Julie King. "We continue to meet and bargain in good faith."

Contract talks stalled in late April with Exxon calling on the USW to submit its proposal to a vote. The union rejected that and submitted its own terms.

Mark Morgan, chairman of USW's negotiating group, said it withdrew an earlier proposal that would have paid control room operators by the hour instead of salary, a move he said would have saved Exxon money.

Exxon has said its offer would allow it to be profitable even during times of low profit margins. It asked for separate agreements for refinery and lubricant workers, and to eliminate parts of the seniority system governing job assignments.

Both have discussed a six-year agreement. The USW offered no pay increase in the first year with the second through sixth years determined by national negotiations that have yet to take place. It wants to continue the seniority system for assignments. (Reporting by Erwin Seba in Beaumont, Texas Editing by Marguerita Choy)