New York Times opinion page editor Andrew Rosenthal insisted Wednesday that his newspaper is not covering up a big story by refusing to release an off-the-record portion of its interview with Donald Trump.

In a Wednesday blog post, Rosenthal told Times public editor Margaret Sullivan that portions of the meeting were meant to be "off the record. But he acknowledged that a recording of the board's talk with Trump was shared with other reporters at the Times, which may have resulted in some details being leaked to the public that were intended to be kept private.

He said, however, that the Times is not covering up any information related to the presidential campaign that would be crucial to the public's knowledge.

"These [editorial board] sessions are not about news gathering for stories," Rosenthal said.

Some people, including Trump's rivals for the Republican presidential nomination Ted Cruz and Marco Rubio, have called on the Times to release the recordings, which reportedly may demonstrate Trump professing "flexibility" on his hardline immigration policy proposals.

Rosenthal has said he would only make the recordings public if Trump asked the paper to release them.

"It's not as if we are sitting on a news story here, or holding something back from publication," Rosenthal said.

Appearing Monday on Fox News, Trump himself acknowledged that at least some parts of his meeting with the Times was off the record. "We had a board meeting, it was off the record, all of a sudden they leak it, it's all over the place," he said. He also said as president, he would be willing to negotiate on immigration policy, of which his trademark policy is to "build a wall" on the Southern border.

"Things are negotiable, I'll be honest with you," he said. "You know, I'll make the wall two feet shorter or something. I mean, everything's negotiate. Building it? Not negotiable."