Two unions representing 65,000 workers at Verizon Communications Inc. have set a Sunday night deadline for contract talks, saying a strike is possible next week.

The phone company’s contract with the Communications Workers of America and the International Brotherhood of Electrical Workers expired last weekend. Their members have been working under the old contract while negotiations have continued, but the unions are stepping up the pressure by setting a deadline for midnight Sunday, just before the workweek starts.

The unions said progress in the talks has been “slow” on important issues like job security and health care.

A Verizon spokesman, Peter Thonis, said the company was “very surprised” by the deadline.

“We and the union leadership have been making consistent progress and have resolved most of the important issues,” he said. “Verizon is committed to continue negotiating as long as it takes to complete the process.”

The union workers are on the traditional telephone side of the business, which has embarked on a labor-intensive project to replace most of its copper phone lines with optical fiber. A strike could mean delays in installations and repairs of phone and broadband lines.

In 2000, at the height of the Internet boom, Verizon workers struck for 18 days before reaching a contract considered a big victory for organized labor. Verizon said then that it would take a month to clear 230,000 requests for repair and unfilled orders that had piled up.

The expiring contract was negotiated in 2003, a month after the previous contract ended. A threatened strike was avoided when federal mediators joined the talks.

The number of union employees at Verizon has declined since the last strike, when it was at 87,000. Verizon Wireless, the company’s big growth driver, has few union employees.