The Tynecastle club hit more than 1,000 on the first day of the briefs going on sale on Tuesday.
Hearts return to the Premiership after a season in the second tier and fans are likely to return for the start of the campaign, albeit it is expected to be in limited numbers to begin with.
More than 16,000 are expected at a Scotland rugby game in July and the Gorgie side hope to follow suit by welcoming back supporters to Tynecastle for the Premier Sports Cup encounters.
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Hearts host Cove Rangers and Inverness CT before the start of the league campaign.
Supporters have been given three options to renew their season tickets with fans able to renew at full price, at a reduced rate due to not being able to attend last season or a ‘Your Hearts Pass’ to secure their seat in future.