The bill would extend relief for an additional 26 weeks and raise the minimum benefit to $135 a week.
"Government funding would fill any gap between the minimum benefit and the amount paid by the absent father," the Times reported.
Under the deal, the minimum weekly benefit, currently $40, would jump to $100.
In 1981, he voted against a move to continue minimum benefit under the Social Security Act.
Mr. Gore would not change the current system that guarantees a minimum benefit.
The extra cost for the minimum benefit has been estimated at 3%, with an additional 2.5% for silver.
"I had to stop working, but then they still give minimum benefits," he said.
The proposal did not specify the minimum benefit, but noted that an amount of about $3,500 for each child had been discussed.
It could also help buy private insurance or a basic package of minimum benefits that insurers would be required to offer in each state.
Connecticut cut costs in June by reducing insurance premiums 17.5 percent and lowering minimum benefits.