“We support the need for groupwide supervision and believe that this supervision is already being carried out by our state regulators, who have deep insight into the insurance industry and our business activities,” the company added in its statement.

Allowing A.I.G. to shed the label could be politically complicated, given the firm’s role in the financial crisis and the $182 billion government bailout it received, although the company has streamlined its operations substantially since 2008.

The Treasury said in July that the financial stability council had convened to discuss issues that included a continuing re-evaluation of the designation status of an unnamed company and the MetLife litigation. The council conducts an annual review of firms that have been designated.

A two-thirds majority of the council’s 10 voting members that includes the Treasury Secretary’s vote is needed to remove a designation. As President Trump appoints additional regulators to top banking posts, the council is expected to become more receptive to removing the label from the remaining firms.

A.I.G. did not respond to requests for comment.

In the 2016 ruling that allowed MetLife to shed the label, Judge Rosemary M. Collyer of Federal District Court in the District of Columbia said regulators had fallen short in their analysis supporting the designation and had failed to properly account for the costs of being designated. The Treasury has been directed to take up those issues in its coming report, which is a response to a White House memo in April that requested a “thorough review” of the designation process.

The court has suspended the MetLife case until the Treasury issues its report.

Critics of the label have often pointed to the financial stability council’s power to scrutinize risky practices as an alternative to singling out individual companies. The theory is that financial activities that could be cause for concern should be monitored industrywide, regardless of a company’s size or market influence.