According to information obtained by News, the surviving bidders for the leading publications are taking part in management presentations in an effort to complete their second-round proposals.
A number of parties, including CEO Anna Jones and Daily Telegraph editor Chris Evans, were set to meet with Telegraph executives, according to people familiar with the process on Tuesday.
They are believed to consist of David Montgomery, a newspaper veteran, and Sir Paul Marshall of National World, a media company registered in London.
It is also believed that at least one other party, whose identity has not yet been made public, is still in the running to purchase the publications.
There are two-way talks between IMI, the Abu Dhabi-based company that sought to acquire a majority stake in the British media assets prior to the government blocking it, and another proposal led by former chancellor Nadhim Zahawi.
According to News exclusively last month, Sir Paul was allegedly pursuing The Spectator, which was owned by the Barclay family along with the Telegraph titles, until their respective holding firms were placed into administration last year. Sir Paul was allegedly chasing The Spectator, which was owned by the Barclay family along with the Telegraph titles, until their respective holding firms were placed into administration last year, as News exclusively revealed last month.