MPs were debating the Prime Minister's Statement on Tuesday when Shane Jones yelled "send the Mexicans home" across the floor.
Winston Peters then told Green MPs Lawrence Xu-Nan and Francisco Hernandez to "show some gratitude" for being in New Zealand.
The Green Party has written to the prime minister and the Speaker of the House calling for action, saying the comments were "outwardly racist and xenophobic".
RNZ approached the Mexican embassy, which confirmed it was "following up this matter through diplomatic channels".
A spokesperson for Peters - who is the deputy prime minister and the foreign minister - said that "on reflection" their comments could have been "expressed differently".
"The deputy prime minister has been made aware of concerns raised by the Mexican Ambassador with MFAT. He looks forward to seeing the Ambassador at Waitangi next week to discuss the matter," the spokesperson said.
"The deputy prime minister is a huge champion of the Mexico-New Zealand relationship, as demonstrated by his visit there late last year. He feels nothing but friendship for the Mexican people.
"In the heat of the moment in the robust environment of Parliament, sometimes some members say things when provoked that, on reflection, may have been expressed differently."
New Zealand First this morning doubled down, with Shane Jones saying the Greens' Mexico-born musterer Ricardo Menéndez March should "grow a pair".
"He swaggers around in Parliament, been there five or 10 minutes and thinks he can tell Winston and I what to do ... he brings alien ideas and woke-ism to New Zealand."
Ahead of Question Time Peters was also unrepentant, saying Parliament was the sort of place where a lot of things were said, and were often "absolutely overreacted to".
He said Mexican Ambassador Alfredo Pérez Bravo had thanked him within the past 48 hours for having visited the country last year, "then I hear this humbug from you guys in the mainstream media, stop wasting everybody's time.
"What's wrong with the rhetoric .. it's an old political dictum, if you can't stand the heat get out of the kitchen."
Prime Minister Christopher Luxon said he would not make the remarks himself, and reminded all MPs to watch their rhetoric - refusing to single out NZ First.
Separately, the ACT Party has also called Te Pāti Māori "race fanatics" after Mariameno Kapa-Kingi said Children's Minister Karen Chhour had a "gap of knowledge" and did not "understand the essence of being Māori".
