By Emily Stephenson

SAN ANTONIO (Reuters) - Donald Trump's rivals for the Republican presidential nomination urged him on Monday to ask the New York Times to release a recording of his recent interview with its editorial board, following a report he told it he was not serious about his immigration proposals.

News website BuzzFeed reported on Monday that Trump had suggested to the Times' editorial board in an off-the-record briefing that he would likely not stand by his immigration proposals if elected president.

Trump, the Republican front-runner, has called for deporting all illegal immigrants and has said he would get the Mexican government to pay for building a wall along the U.S.-Mexico border.

"Donald Trump should ask The New York Times to release the audio of his interview with him so we can see exactly what it is he truly believes about this issue that he has made the cornerstone of his campaign," U.S. Senator Marco Rubio of Florida said in a statement.

U.S. Senator Ted Cruz of Texas called on Trump to do so before this week's Super Tuesday, the biggest single day of state-by-state presidential nominating contests for the Nov. 8 presidential election.

"There's one of two instances: It is either false ... (or) he actually now is telling the New York Times editorial board: 'Pay no attention to what I'm saying on immigration because I, Donald Trump don't intend to do anything I'm saying,'" Cruz said at a campaign rally in San Antonio.

BuzzFeed quoted New York Times editorial page editor Andrew Rosenthal as saying he would not comment "on what was off the record at our meeting with him."

"If (Trump) wants to call up and ask us to release this transcript, he’s free to do that and then we can decide what we would do," BuzzFeed quoted Rosenthal as saying.

New York Times spokeswoman Danielle Rhoades Ha said the newspaper was "not commenting beyond what our editorial page editor told BuzzFeed."

A Trump official was not immediately available to comment.

(Reporting by Emily Stephenson in San Antonio; Additional reporting by Steve Holland and Richard Valdmanis; Editing by Peter Cooney)

This article was funded in part by SAP. It was independently created by the Reuters editorial staff. SAP had no editorial involvement in its creation or production.