Last month, the groups called on The New York Times to invest more heavily in digital and to sell some other assets to finance these acquisitions.
The paper, which has been identified as a possible takeover target in the future by the likes of Google, was being lobbied by hedge funds Firebrand Partners and Harbinger Capital Partners.
The two companies previously owned 4.9% of the paper but have now increased their stake, according to an SEC filing. Scott Galloway, head of Firebrand, has met with Times chairman Arthur Sulzberger Jr. and CEO Janet Robinson.
The New York Times has made no response as of yet, but it is expected to oppose the hedge fund's bid for four of the 13 seats on the board, which are elected by public shareholders rather than the controlling Sulzberger family.
The hedge funds have not as yet taken a hostile approach to the New York Times and have called for a change in the voting structure to the class B shares, which gives the Sulzberger family control of the company.
In a letter to Galloway, Sulzberger and Robinson did not disclose what was discussed, but said: "We are looking forward to continuing a productive and positive dialogue.
"Accordingly, as we discussed on Friday, each of our director nominees is free to meet with the members of your nominating committee at their convenience."
Firebrand-Harbinger has proposed four nominees for the NYT board, including Galloway, Firebrand adviser; Gregory Shove, former AOL executive; Allen Morgan, Mayfield Fund managing director; and James Kohlberg, co-founder of private-equity firm Kohlberg & Co.
The bid to change the company's strategy follows an attempt by another shareholder, Morgan Stanley, a year ago. The investment bank sold its stock after the bid failed.