The Boston Teachers Union and the School Department on Friday inched toward resuming talks over a new contract, as both sides expressed a willingness to meet next week.

The two sides could sit down on Tuesday, a session that both characterized as tentative. It would represent the first face-to-face meeting of the two parties since mid-August.

The development occurred as the teachers union sent the School Department a proposal late Friday afternoon that countered a School Department proposal issued earlier this week. That School Department proposal was in response to a previous one by the union.

The two sides have been negotiating a contract for more than two years. “We are willing, ready and able to talk,” said Richard Stutman, the teachers union president.

However, Stutman added that he still needs to check with the union’s negotiating team to make sure they could attend a Tuesday meeting. Matthew Wilder, a School Department spokesman, said school officials would first review the union’s counterproposal before deciding whether to meet Tuesday. “I think we are very willing to sit down,” he said.

The two sides have been trading proposals as the state Department of Labor Relations prepares to launch an investigation into the contract stalemate and recommend a resolution, at the request of Mayor Thomas M. Menino.

Sticking points have been wage increases and the creation of a new system to evaluate teachers. Last week, the union said it would accept the city’s wage offer and proposed that the two sides use a state-developed process to evaluate teachers, which the School Department agreed to this week.

But the union’s acceptance came with conditions that the School Department accept certain proposals, such as reducing class sizes at certain grade levels. The School Department did not agree to that proposal and a few others this week.

At the same time, the department asked that the union agree to some of its proposals, such as granting pay raises to new teachers based on performance rather than guaranteeing them annual increases.

The union’s proposal Friday night counters some of the School Department’s items and includes some new items that the union contends the two sides previously agreed to.