Around 80% of Mexico's population is thought to be Catholic, one of the world's largest Roman Catholic nations.
Enrique Pena Nieto has not taken a stance on the issue previously and did not speak about it during his election in 2010, now he said he will put it into law.
He was speaking on Tuesday, the International Day Against Homophobia.
In 2015 the Supreme Court in Mexico ruled that it was unconstitutional for states to bar same-sex couples from marrying.
The Daily Mail has reported that Pena Nieto said he wants to change Article 4 of the constitution so that it reflects the Supreme Court view "to recognise as a human right that people can enter into marriage without any kind of discrimination."
He went on: "That is, for marriages to be carried out without discrimination on the basis of ethnicity or nationality, of disabilities, of social or health conditions, of religion, of gender or sexual preference."
Por un @Mexico incluyente que reconoce en la diversidad, una de sus mayores fortalezas #SinHomofobia. pic.twitter.com/6ooZGuPEsD
— Enrique Peña Nieto (@EPN) May 17, 2016
Lesbian, gay, bisexual and transgender (LGBT) activists have voiced support for the president.
To amend the constitution Mr Nieto needs a two-thirds majority in congress.
His party and allies control around half of the seats in both houses.
Not everyone is happy about the president's change of heart, Revd Hugo Valdemar, spokesperson for the Catholic Archdiocese of Mexico City, has urged lawmakers oppose the idea.
He said: "Marriage has some very concrete aims which, of course, two people of the same sex do not fulfil."
Revd Hugo Valdemar said this was a ploy to distract from more serious issues like violence and corruption.
