LATEST UPDATE as of 7:35 p.m.: Negotiations between the school bus drivers' union and Durham School Services will resume tonight and likely will continue on Wednesday.

L.D. Fletcher, president of the local drivers' union, said that even if a deal is not reached Tuesday night, the union would not strike on Wednesday.

Charleston County School Superintendent Nancy McGinley said she has been assured that both sides want to sincerely come to an agreement and will exhaust every avenue to avoid a work stoppage, including consideration for mediation.

School buses in Charleston and Dorchester 2 will run their regular morning and afternoon routes on Wednesday.

LATEST UPDATE as of 2:35 p.m.: Durham School Services says it did not walk out on meetings with the bus drivers' union, and it remains committed to reaching an agreement that benefits everyone.

The company that employs bus drivers in Charleston and Dorchester 2 said it increased its offer but still can't meet the “unreasonable demands of the union.”

“While the union rejected our offer of mediation made earlier in negotiations, we stand ready to continue discussions, whether through continued negotiations or mediation, should they decide to adjust their demands,” said Carina Noble, spokeswoman for Durham, in a statement.

EARLIER IN THE DAY:

Durham School Services officials walked out of a meeting with the school bus drivers' union Tuesday morning after the union refused the company's employment package offer, according to the president of Teamsters Local 509.

L.D. Fletcher, president of the local Teamsters union, said the union still is willing to talk further with the company, and they're not calling a strike today.

Fletcher said school buses will run their regular afternoon routes today.

Fletcher planned to visit school superintendents in Dorchester 2 and Beaufort to update them on the status of negotiations. The union has been trying to work out a new employment contract with the company since last July, and contract talks have heated up during the last two weeks.

The union has promised not to strike while talks are ongoing.

Check back later for more information, and follow @Diette or @postandcourier for the latest updates provided to media.