Finance minister of France, Christine Lagarde, is favourite to take the job.
However, Mr Carstens may hope to benefit from a backlash among developing countries against Europe's historic domination of the position.
The previous head - Frenchman Dominique Strauss-Kahn - quit last week to fight sexual assault charges in New York.
The Mexicans' announcement on Sunday came a day ahead of the formal opening of nominations, with a deadline of 10 June set by the IMF for all candidates to be put forwards.
A shortlist of three will then be drawn up over the following week, from which the Fund's executive board will choose the new head by 30 June.
Mr Strauss-Kahn was the fourth Frenchman to have held the IMF's top job.
He is in New York on bail awaiting trial for sexual assault and attempted rape, charges he denies.
In their statement Mr Swan and Mr Gordhan said G20 leaders wanted to reforming international financial institutions, including the IMF and the World Bank.
Developed industrialised countries currently control about 57% of the voting rights in the IMF, compared with about 43% for developing nations.
With the recent emergence of Brazil, Russia, India and China, many other global figures - including the head of the Organisation for Economic Co-operation and Development - had also called for the IMF job to go to a non-European.
The appointment is usually decided unanimously by the IMF's executive board, although a majority vote by the board would be enough to choose the winner.
