Information about 2,743 individuals was sent to 77 people in an email attachment by mistake last month.
The authority has apologised and the member of staff involved has resigned.
Sunderland-based law firm Ben Hoare Bell said it was seeking compensation for an undisclosed number of families that had suffered "real distress".
The details, dating back to 2004, included names, addresses and birthdates of the adopted children as well as information about adoptive parents and social workers.
The Information Commissioner's Office is investigating and has the power to levy a fine of up to Â£500,000.
Solicitor Richard Hardy, who specialises in civil liberties cases, said: "The breach has caused real distress to those whose details were disclosed in the spreadsheet.
"The fact a child has been adopted usually means there is a complex and difficult background to the child's life.
"To hear highly sensitive information including old and new names, dates of birth and addresses have been released into the public domain has caused substantial anxiety and distress.
"There is a fear this information will make its way onto social media and significantly impact those concerned. It can only be hoped the council's efforts to encourage the recipients to delete the data unread will be wholly successful."
A council spokesman said: "We have reported this to the Information Commissioner's Office and cannot comment any further."
Newcastle City Council is facing legal action after the details of thousands of children and their adoptive parents were leaked in a party invitation.