The Australian Rugby Union said that general public ticket allocations for Tests in Brisbane, Melbourne and Sydney were "drained" within minutes of going on public sale on Monday.
The Brisbane allocation was snapped up inside five minutes, and it took less than 15 minutes for available tickets to go in terms of the Melbourne and Sydney games.
Interest has not been witnessed on such a scale Down Under since the 2003 World Cup, which was hosted by Australia and saw England crowned champions.
Tickets are still available for the Lions' fixtures against all five Super Rugby teams - Western Force, the Reds, Waratahs, Brumbies and Melbourne Rebels - plus a midweek appointment with a Combined NSW-Queensland Country side.
The 10-game Lions tour kicks off against the Barbarians in Hong Kong on June 1, with the Tests scheduled for June 22, 29 and July 6.
The Lions have not won a Test series since they clinched a 2-1 verdict against South Africa in 1997.