Chile, who won the Copa America for the first time last year, have been looking to replace their title-winning coach Jorge Sampaoli since he quit 10 days ago.
"An agreement has been reached for (Pizzi) to be the next trainer of the Chilean national team," ANFP vice-president Andres Fazio told reporters.
"It's most probable that we'll have him here one day next week."
Pizzi, who has been coaching Mexico's Leon since December 2014, will have less than two months to prepare for Chile's next two qualifiers at home to Argentina and away to Venezuela towards the end of March.
The Argentine-born 47-year-old has worked in Chile before between 2009 and 2011, steering Universidad Catolica to the league title. He also won the Argentine Inicial championship with San Lorenzo in 2013.
Pizzi, speaking at a news conference in Mexico earlier on Friday ahead of Leon's Liga MX visit to champions UANL Tigres on Saturday, said he was honoured to have been approached by Chile.
Fazio said the ANFP would not be paying Leon any compensation.
Argentine Sampaoli left the Chile job after details of his contract with the ANFP were leaked to the media.
Chile are fifth in the 10-team South American qualifying competition for the next World Cup in Russia, having picked up seven points from four matches. The top four gain automatic qualification with the fifth-placed side entering a playoff.
(Reporting by Javier Leira; Writing by Rex Gowar; Editing by John O'Brien)
