NZRU deputy chief executive Steve Tew revealed that 16 per cent of the tickets available to New Zealanders for the series had yet to be sold.
"We will be going out to all people in the ballot with a final offer before going to public sale on May 2," he said.
"As expected, we have seen huge demand for tickets to the Tests and major matches and while demand has been less strong for some of the provincial matches, we expect the local demand to ramp up when we go to public sale.
"The Tests are sold out and there has also been strong demand for the New Zealand Maori game as well as the Auckland and Wellington games."
Once the final offer to those in the ballot has closed the remaining tickets will go on general sale, meaning Lions fans will also now be able to purchase tickets to those games.
The Lions' 11-match tour, featuring three Tests, begins against Bay of Plenty in Rotorua on June 4.