Clarke was ruled out of the rest of the France series on the eve of the second match in Wellington on Saturday, but All Blacks coach Scott Robertson said he could miss more Tests.
"He could be probably six to eight (weeks). It is a good ankle injury," Robertson said on Monday.
"Shame for him, really gutted."
The All Blacks play their first two Rugby Championship matches against Argentina in Cordoba and Buenos Aires on August 16 and 23.
Clarke was replaced in Robertson's starting 15 in Wellington by Emoni Narawi. The All Blacks thrashed France 43-17 to seal the series and make the third match in Hamilton on Saturday a dead rubber.
Clarke was originally named in the second Test team as a replacement for the concussed Sevu Reece. The Fiji-born winger is now available for Hamilton after clearing concussion protocols.
However, Robertson will be without Tupou Vaa'i, after the lock-turned-flanker suffered a head knock in the second half in Wellington and later failed a concussion test.
Vaa'i, who started in the No.6 shirt in the first two Tests in a rejigged back row, had been in good form, scoring a try in each match.
Loose forward Luke Jacobson will be a potential replacement after recovering from injury, while prop Tyrel Lomax is also available after missing the first two Tests.
With the series sewn up, Robertson has the luxury of giving fringe players a run, and mooted picking anyone fit in his 33-man squad who were overlooked for the first Tests.
That could mean game time for prop George Bower, hooker Brodie McAlister, halfback Noah Hotham, fullback Ruben Love and veteran midfielder Anton Lienert-Brown.
"There's some competition in the team. There's competition all over, which is exactly what we want," Robertson said of the World Cup runners-up.
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