With football postponed until at least April, owner Ann Budge has proposed the cost-cutting measure in an attempt to keep the club afloat should the virus delay the domestic season any further.
Hearts currently sit bottom of the league, seven points from safety, and are the first top-flight team to take such measures after Budge acknowledged an impending loss of £1 million ($1.16 million) in deferred fixtures.
"We have asked all full-time employees, managers, coaches, players and player back-room staff, with effect from the beginning of April, to accept a 50 per cent cut in their monthly salary," Budge told Hearts' official website.
"Staff and players alike, who feel unable or unwilling to accept this revision to their contracts, will, of course, be offered the option of contract termination."
Socceroos midfielder Bozanic first signed with the club in June 2018 and has since gone on to make 60 appearances for the club, alongside seven goals and two assists.
Garuccio also joined in 2018 after leaving Adelaide United. The 24-year-old left-back has played 29 games for the club.
Though the wage cut would come into effect from the beginning of April, the club insists the players will not receive less than the United Kingdom's National Living Wage.
"Given the uncertainty of the whole situation with which we have been presented, we cannot say how long these measures will be in place," she added.
"This reduction in income is not sustainable without taking immediate action to cut staff costs and overheads."
Budge's decision came on the same day Scottish Football Association president Rod Petrie announced an immediate £1.5 million (A$3.1 million) fund to help clubs and academies in need.