The titles are The Times Educational Supplement, The Times Higher Educational Supplement, as well as smaller titles such as Nursery World. However, The Times Literary Supplement is not part of the sale.
The Financial Times has reported that Apax Partners, Candover and Exponent have gone through to a second round of the auction, with final bids due in the first week of September.
Analysts estimate that the bids will start at £230m and go as high as £280m.
The companies that declined to take their bids further were private equity companies Kohlberg Kravis Roberts and Carlyle Group, and publishers including Emap and Guardian Media Group.
Some of the publishers questioned whether the sale might have been prompted by competition from online jobs advertising, which is thought to have hit revenue streams at the papers.
However, in its 2004 results News Corp said that classified advertising at the flagship The Times Educational Supplement was higher as a result of a strong recruitment sector and The Times Higher Education Supplement and Nursery World all reported an increase in circulation.
The titles and their associated exhibitions are part of News Corp's TSL business, formerly known as Times Supplements Limited until its name was changed in 1999 to avoid confusion over the company's relationship with The Times newspaper.
The 2004 figures for TSL indicate it was in good shape, earning pre-tax profits of £23m on revenues of £56.1m.
News Corp has not disclosed its reasons for selling off the titles, but is believed to need money for investment in its printing plants.
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