Bidders for The Daily Telegraph and its Sunday edition are engaging in direct talks with the newspaper group's management this week, ahead of a formal bid deadline later this month, according to Sky.
The remaining bidders are participating in presentations with key executives, including CEO Anna Jones and editor Chris Evans, as they finalise their second-round offers.
Interest has come from Sir Paul Marshall and National World, led by David Montgomery. Another unidentified party is also in the running, according to Sky.
A separate bid led by former Chancellor Nadhim Zahawi, involving Abu Dhabi's IMI, is also in bilateral discussions.
The sale follows the liquidation of the Barclay family's holding companies last year, which left the fate of the Telegraph titles uncertain. RedBird IMI, a joint venture, previously attempted to acquire the titles but was blocked by new media laws.