In his conversation with Nieto, according to the transcripts, Trump is heard pleading that the latter should stop saying that Mexico would not pay for the wall.
In his conversation with the Australian Prime Minister, Trump is heard getting agitated on the refugee issue and eventually telling him that the call was the most irksome of the day.
Transcripts of both the conversations that happened on January 27 and 28 respectively were posted by The Washington Post.
The White House today did not immediately comment on the embarrassing leak of Trump's conversations with world leaders. In a statement, The Washington Post said the transcripts were prepared by the White House but have not been released.
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"The Post is publishing reproductions rather than original documents in order to protect sources. The reproductions below also include minor spelling and grammatical mistakes that appeared in the documents," it said.
The building of the wall was one of the major electoral promises of the US President. During his election campaign, Trump had said that Mexico would pay for the wall. Pena Nieto has repeatedly said that Mexico will not pay for the wall.
"You cannot say that to the press. The press is going to go with that and I cannot live with that. You cannot say that to the press because I cannot negotiate under those circumstances," Trump told Pena Nieto, according to the transcripts.
"We have to stop. We have allowed so many people into our country that should not be here. We have our San Bernardino's, we have had the World Trade Center come down because of people that should not have been in our country, and now we are supposed to take 2,000. It sends such a bad signal. You have no idea. It is such a bad thing," Trump is quoted as saying in the transcripts.
The conversation between the two leaders grew sour as Trump rejected an agreement to take refugees.