The heavily indebted British travel group put its profitable airline business up for sale in February after profit warnings in 2018 left it needing to raise cash.
Thomas Cook's airlines business consists of Germany's Condor, as well as British, Scandinavian and Spanish operations.
A sale, in whole or in part, would enable the world's oldest tour operator to pay down debt and invest more in its own hotels and improve its online sales in a bid to differentiate its offering from rivals.
A source familiar with the discussions said that Indigo and Germany's Lufthansa were most interested in the business.
British Airways owner IAG should not be ruled out and easyJet has engaged in talks but is seen as less interested, the source added.
It is not clear whether Ireland's Ryanair would bid.
Another source said that private equity groups KKR and Apollo might also look at taking over the whole of Thomas Cook.
Thomas Cook, Indigo, IAG, easyJet and Ryanair declined to comment. Lufthansa had no immediate comment.
The airlines business provides access to valuable European slots linking Britain and Germany to key holiday destinations. However analysts at RBC Capital Markets said the airline fleet was ageing and some of the slots had limited value.
Shares in Thomas Cook fell 4.3 percent to 24.82 pence per share by 1048 GMT, underperforming a flat European transport & leisure index.
Lufthansa executives have said repeatedly that the German airline wants to "play an active role" in consolidation.
Indigo, the private equity firm managed by veteran U.S. low-cost airline investor Bill Franke, has previously made investments in several airlines including Hungary's Wizz .
Analysts at Goodbody said the most likely initial outcome was the sale of the Condor long-haul business to Lufthansa, but that they were concerned about what would happen to the competitive short-haul unit.
"Depending on how this sale process plays out, this adds to our concern on penetration of German-EU flows from potential low-cost carrier competition," they said.
Thomas Cook has been revamping various parts of its business this year, closing high street stores and reviewing its money division as it focuses on holidays.
The company was hit badly in 2018 when a hot European summer deterred customers from booking holidays through the year.
One banking source said the airline would fetch less than 1 billion euros ($1.12 billion), including debt. Thomas Cook has a current market value of around 400 million pounds ($523 million).
Sources said that competition issues could influence which parts of the business different suitors go for.
Sky News has said China's Fosun International, a Thomas Cook shareholder, was interested in its tour business.
(Reporting by Kate Holton and Clara Denina in London; additional reporting by Alistair Smout and Georgina Prodhan in London and Arno Schuetze and Edward Taylor in Frankfurt; Editing by Alexander Smith and Georgina Prodhan)
By Kate Holton and Clara Denina