The union sent out mail ballots to the 8,479 delegates who attended its last convention and got responses from about 6,200.
The union's members will vote by mail ballot later this month.
The winner needs a majority of the votes cast in the mail ballot.
He said it could take as much as six weeks for the union's membership to vote on the agreement by a mail ballot.
Union officials said that about 20,000 workers have voted since the mail ballots went out three weeks ago.
Perhaps most important, given the concerns about voting machine security, mail ballots cannot be hacked.
The unions will vote on the agreement by mail ballot within the next month.
A mail ballot will then be sent to the organization's 119-member board.
In a mail ballot, however, the bishops voted 150 to 98 to reject the translation.
The mail ballots, submitted over the last two weeks, were counted yesterday.