The U.S. top financial risk council may tweak the way in which it identifies insurers and other companies that are not banks as systemically important, a process the industry and politicians have fiercely criticized.

The Financial Stability Oversight Council, a group of the nation’s main financial regulators, has named three such companies as being so important that their demise could pose risks to the global system, subjecting them to tougher oversight.

“The council has received a number of suggestions regarding its process for evaluating non-bank financial companies for potential designation,” Treasury Secretary Jack Lew said at a meeting of the FSOC, which he chairs. “The council will begin to examine possible changes in the coming months.”

Lew said that the council had received suggestions on how to improve its process around the designations and that he had asked staff to continue to talk to the industry and then report back to him to see if changes were needed.

Insurer MetLife Inc on Friday said it would fight a plan by the FSOC to designate it as a so-called SystemicallyImportant Financial Institution (SIFI), which would make it the fourth non-bank firm overseen by the Federal Reserve.

The 2010 Dodd-Frank Wall Street reform act defined banks with more than $50 billion in assets as “systemic” but left it to FSOC to determine whether some non-banks also deserved the tag.

The FSOC has so far designated insurers American International Group Inc and Prudential Financial Inc as SIFIs, along with GE Capital, the financial services arm of General Electric Co.

The designations are part of an effort to rein in the largest and most risky companies after the collapse of investment bank Lehman Brothers during the credit crisis showed how trouble at just one company could roil global markets.

The $182 billion bailout of AIG after it had racked up huge losses from risky financial bets showed that not just banks can bring the financial system to the brink.

FSOC will now hold a hearing where MetLife can make its case and must then make a final decision within 60 days, which the insurer could challenge in court.

Separately on Monday, minutes released from an FSOC closed-door vote on Metlife showed that the two members who are insurance experts have concerns. Roy Woodall, an independent voting member, declined to cast a yes or no vote. Another insurance regulator, who does not have a vote, also “expressed several areas of concern” about the designation.

Patrick Pinschmidt, the FSOC’s deputy assistant secretary, said last month the council wanted to consider additional reforms, speaking at a congressional hearing after the Government Accountability Office faulted the council for a lack of transparency and other flaws.

Both Democratic and Republican lawmakers have urged the FSOC to be less secretive and involve the industry more into its decisions to name insurers as systemically important.

Critics are also upset because regulators have not yet written rules that spell out what new regulations will apply to designated companies, leaving them in limbo. (Reporting by Douwe Miedema; Editing by Steve Orlofsky)