The Indian government had tried to offload its stake in Air India in 2018 but failed to attract a single bid.
One group is representing employees and plans to offer them a controlling stake in the struggling airline.
Another bid is reported to have been put forward by the Tata Group, which originally founded the airline in 1932.
Tata, which owns Jaguar Land Rover, sold its stake to the government in the 1950s.
India's Prime Minister Narendra Modi is keen to sell the government's entire interest in the airline, which has been kept aloft by a bailout and racked up billions in debts.
The airline has many assets, including prized slots at London's Heathrow airport, a fleet of more than 100 planes and thousands of trained pilots and crew.
One of the bids put in ahead of this week's deadline was from US-based investment firm, Interups.
Under its plan, Interups will hold 49% of Air India while a controlling stake of 51% will be held by its employees.
"We are giving an open offer to employees of Air India to substantially own the airline," Interups chairman Laxmi Prasad told the BBC.
"Our group will invest the entire monies required for the airline, with no capital requirement from employees to contribute into the acquisition effort."
Calling them the "backbone to run the airline", Mr Prasad added that the 51% stake would be "in exchange for the deep intangible contribution you all would be making for the airline."
"No-one knows Air India better than its employees and management."