Cheika worked with the Pumas for the last two years as an assistant to former head coach Mario Ledesma, who stepped down last month after a string of poor results.
The two have had a long history of working together with Ledesma serving as scrum coach under the Australian at the Wallabies.
Cheika, 55, beat out former Pumas playmaker Gonzalo Quesada, the head coach of French club Stade Francais, to replace Ledesma, La Nacion reported.
Leinster assistant coach Felipe Contepomi will join Cheika's staff with the intention of succeeding him after next year's World Cup in France, the newspaper added.
Reuters was not able to immediately reach Cheika and the Argentine Rugby Union for comment.
A championship-winning coach with Irish club Leinster and Sydney-based Super Rugby side NSW Waratahs, Cheika guided the Wallabies to the final of the 2015 World Cup in Britain the year after taking over the team.
He resigned in acrimony four years later, however, after Australia crashed out of the World Cup quarter-finals in Japan, matching their worst result at the global tournament.
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