The company said on Tuesday that it has started to send out letters to the strikers in response to questions about how healthcare coverage will be affected by the strike, which started on August 7.
The Communications Workers of America (CWA) and the International Brotherhood of Electrical Workers (IBEW) called a strike after they failed to reach a labor agreement with the company after their contract expired.
Their disagreements hinge on areas such as healthcare contributions, pensions and work rules, according to both sides.
CWA spokeswoman Candice Johnson said the union would set up a plan and make sure striking workers' healthcare needs are met if the workers are still striking by August 31.
Both Verizon and CWA said that talks with the company were still going on but neither side was able to say whether they had an agreement in sight.
Verizon spokesman Rich Young said the company had made some progress on certain issues and some issues are still up for discussion. He did not give details.
If they are still on strike the workers, which include technicians and customer service representatives, could pay for their current health insurance plans, he said.
This would cost them about $4,800 to $10,000 per year for individual plans or $10,000 to $20,000 for family plans, Young said.
(Reporting by Sinead Carew, editing by Bernard Orr)