Mexico are looking for a manager to take them to the 2010 World Cup finals in South Africa.
Eriksson will not officially take up his post until 21 June, when Mexico host Belize Monterrey in the second leg of the second round of World Cup qualifiers.
Until then caretaker coach Jesus Ramirez will remain in temporary charge for the matches against Argentina on 4 June, Peru on 8 June and the first leg with Belize in Houston on 15 June.
Eriksson is Mexico's first foreign coach since Serbian Bora Milutinovic, who resigned in 1997 from his second stint in charge of the team.
Apart from Milutinovic, Mexico has had two Spaniards, two Hungarians and an Englishman as coaches.
Eriksson's season-long stint at City was officially brought to an end on Monday by the club's owner Thaksin Shinawatra, following weeks of speculation.
BBC Sport revealed in April that Eriksson was no longer wanted by owner Thaksin, despite City finishing ninth in the Premiership.
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