The teachers, from schools in Manhattan, the Bronx, Staten Island and Westchester County, are alleging unfair labor practices.
McEvilly said the archdiocese hasn't provided union negotiators with information it needs about health insurance.
"We asked for this information more than three months ago. As a result we couldn't bargain adequately for our people because we wanted to present them with an adequate health care proposal," McEvilly said.
Archdiocese spokesman Joe Zwilling had said earlier that the archdiocese has given the union all the data it requested.
McEvilly said the two sides last met in January and have a session scheduled for next Wednesday.
Asked if the union planned another job action, McEvilly said it would depend on whether the archdiocese provided the requested material.
The Federation of Catholic Teachers filed an unfair labor practice charge against the Archdiocese's Association of Catholic Schools (ACS) at the New York State Employees Relations Board (NYSERB) on Thursday.
In letters sent home with students, the archdiocese has told parents that it is offering substantial wage increases to the teachers. But the union contends its teachers are are among the lowest paid professionals in the northeast and earn half the salaries public school teachers earn.
Union officials say the archdiocese has failed to tell parents it is also demanding steep increases in payments by the teachers for medical coverage.
The union has been negotiating with the ACS for almost a year, and its members have been working without a contract since September 1, 2007.
The union says that faced with the ACS's demands for steep increases in the cost of medical coverage, union leaders say they suggested a shift to another health care plan. To see if this compromise was viable, they requested specific information from the ACS more than 100 days ago.
They say the information has never been provided, prompting the filing of the unfair labor practice charge and the beginning of a strike, which took place in the form of sickouts at 10 schools in the Archdiocese.
The affected schools were located in Manhattan, the Bronx, Staten Island, Hartsdale, Scarsdale and New Rochelle.