Potential suitors to buy the Telegraph Media Group have until the end of Friday to submit second-round offers.
It is understood at least three parties, including both UK and international firms, are in the frame to buy the business.
It is the latest stage of a twisting ownership process for the historic London-based publisher.
The company, and fellow publication The Spectator, were originally put on the market last year in order to help previous owners, the Barclay family, repay heavy debts to lenders at banking firm Lloyds.
Following this process, Abu-Dhabi backed media investor RedBird IMI took control of the Telegraph Media Group and the Spectator.
RedBird IMI is an investment fund majority-owned by Sheikh Mansour bin Zayed Al Nahyan, vice president of the United Arab Emirates (UAE) and owner of Manchester City Football Club.
It is also backed by US investment management firm RedBird Capital and is led by former CNN boss Jeff Zucker.
The deal drew fierce criticism over free speech concerns before it was ultimately blocked by the Government earlier this year.
As a result, RedBird IMI launched a fresh sale process to sell on the titles.
Earlier this month, it sold The Spectator to hedge fund tycoon Sir Paul Marshall’s OQS Ventures business for £100 million.
RedBird IMI whittled down potential interested parties to buy the Telegraph earlier in the summer after an initial deadline for indicative offers.
It is understood that Sir Paul Marshall is still among those involved in the auction process to buy the Telegraph, with financial support from US billionaire Ken Griffin.
Yorkshire Post owner National World is also understood to be in the process to strike a takeover deal.
The British-born owner of the New York Sun has also tabled a bid to take control of the media business.
It is understood that Dovid Efune, who is also former editor of Jewish publication the Algemeiner Journal, is able to offer a competitive bid of about £550 million.
Potential rival bidders also include former chancellor Nadhim Zahawi, who approached a number of potential financiers in recent months to assemble a bid.
Private equity firm CVC, which owns sports brands including the Six Nations and La Liga, has also held talks about a potential offer, but it is not know if it remains in the process.