The changes mean that if the federal mandate to buy insurance goes away - some consumer protections in state law would expire a year later.
Those protections include a guarantee that people with pre-existing conditions can buy coverage. They also dictate how insurance companies can set premiums.
Democratic Assemblymember Richard Pan says the rules would give state lawmakers time to adopt alternatives.
DR. PAN: "We wanted to be sure that we didn't just take the protections away immediately and that the legislature would have the chance to try to solve the problem if for some reason the federal government did change the law. But that's a big 'if.'"
Last year, the Governor vetoed a similar bill because he said the consumer protections need to be linked to the federal requirement to buy insurance.
Consumer protections alone, he wrote in his veto message, "would lead to skyrocketing premiums, making coverage more unaffordable."
Other changes to the bill this week include increasing the number of geographic rating regions that insurance companies can use to price premiums.
The bills are expected to be voted on in the Health Committees of both state houses on Tuesday.