The company sued Communications Workers of America leaders Wednesday in New York and got a court order Monday in Pennsylvania.
A Verizon Communication Inc. spokesman says the company got a similar order Wednesday in Delaware and has legal action going in New Jersey and Massachusetts.
A union spokeswoman didn't immediately return a phone call Wednesday. The union has said it doesn't condone any illegal activity.
Some 45,000 Verizon landline workers struck Sunday from Massachusetts to Virginia.
The New York-based company says managers have been filling in for the striking workers.