The others that had lodged proposals included BGH Capital, Indigo Partners and Brookfield Asset Management, Reuters had previously reported.
Virgin owed nearly A$7 billion ($4.76 billion) to creditors when it entered voluntary administration in April with its long-running financial struggles exacerbated by the coronavirus pandemic.
The next phase for the parties on the final shortlist will include further engagement with stakeholders and aircraft financiers as they seek agreements on future terms before binding bids are received, Deloitte has said.
The strong interest in Virgin at a time when the world aviation market is largely grounded shows the long-time attractiveness of the Australian domestic market, a duopoly between Qantas Airways Ltd (QAN.AX) and Virgin.
Bain, which owns Trans Maldivian Airways, is being advised on its Virgin offer by Jayne Hrdlicka, the former head of Qantas budget airline Jetstar.
Cyrus was an investor in Virgin America alongside Virgin founder Richard Branson before its sale to Alaska Airlines. Cyrus also invested in collapsed British carrier Flybe with Virgin Atlantic.