In March, the House voted to ban the procedure by a veto-proof margin.
By a veto-proof margin in both chambers, they passed a $14.3 billion budget that called for no new taxes.
Democratic support for the legislation is strong enough that the House has already passed it by a veto-proof margin, last month.
Both chambers, however, approved the bill by veto-proof margins.
The House already passed similar legislation, but by 39 votes short of a veto-proof margin.
A day earlier, the House failed to get a veto-proof margin on the same measure, voting 272 to 156, 14 votes short.
Still, he expressed doubt that the House could pass the Senate version by a veto-proof margin.
An earlier version of the bill passed the House last year by a veto-proof margin of 313 to 108.
In the Senate, Republicans will have a 24-16 majority, narrowed from its veto-proof 27-13 margin of the last two years.
House and Senate supporters of the plan said they are unlikely to have veto-proof margins.