The House has already passed the same ban by a veto-proof majority.
"Two weeks ago," one senior Administration official said, "we thought this bill might pass with a veto-proof majority."
The Democrats hold a veto-proof majority in both houses of the legislature.
He predicted the House would be "in range" of winning a veto-proof majority.
And if they did win only five, Republicans would have a veto-proof majority.
Democrats now have a veto-proof majority in the State Senate.
That is just two seats short of the 101 needed for a veto-proof majority.
In neither house was the bill passed by a veto-proof majority.
The House passed a similar measure by a veto-proof majority.
The House is also expected to take up the bill next week, but its prospects for a veto-proof majority are uncertain.