The BBA is proposing a widespread advertising campaign to ensure people are aware of the PPI issue and know how to complain before a deadline of April 2014.
The FSA said it would hold a full public consultation before any decisions were made.
A statement from the FSA said: "Our key priority is to ensure that consumers are protected, so the FSA board would need to be convinced that any proposals would be in the interests of consumers.
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"We have had initial discussions and are prepared to consider the merits of this and other options. A key consideration will be the potential to get compensation to more consumers, more quickly.
"We will continue to hold discussions with the BBA as well as actively seeking the opinions of consumer groups and other stakeholder. However, no changes to existing FSA or future Financial Conduct Authority rules would take place without a full public consultation."
Figures from the Financial Ombudsman Service earlier this month showed it dealt with 5,000 PPI complaints every week and expects to receive 250,000 complaints by the end of 2012/13.
Some £13bn has already been set aside by banks and building societies in compensation and more than 2.5m people have claimed for mis-sold insurance.

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