The Scotland skipper will sit out Saturday's encounter with the Pumas in Tucuman but could return in time for the second Test on June 19.
"I didn't train fully last week and I probably won't train fully this week until Thursday or Friday so it's too late for the first Test," said Cusiter.
"The injury is improving, slightly, but improving. I have to keep working hard with the medics."
"Chris is out of the first match," added Scotland head coach Andy Robinson.
"I will announce the captain on Thursday, alongside the rest of the line-up."
Despite the setback, Scotland go into the series in good spirits having ended Ireland's long unbeaten run at Croke Park in the final round of this year's Six Nations.
And Morrison believes that even the inconsistent nature of the 24-5 warm-up win over the Japanese Select side will have positive effects come Saturday afternoon.
"It definitely helped to have that game last week," said Morrison, who is expected to start the first Test at inside centre.
"We hadn't been together as a squad since we played Ireland in the Six Nations. Rather than just train against each other, to play a competitive game against an unknown entity definitely helped us.
"It made us realise that there is a lot of work to do in preparation for this game.
"We won, and that's what we set out to do, but we realise we could have been far more clinical in several areas of the game.
"That's what we've been addressing this week and we'll take that as our gee-up and hopefully go into these games in the best form we can and win them both."
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