Messi was suffering with a sore throat and a fever prior to Argentina's 2-0 win over Chile on Wednesday, but he then tweaked his right hamstring in the second half before managing to see out the full 90 minutes.
The 36-year-old has acknowledged that the issue is "nothing serious" but he did "find it difficult to move freely" towards the end of the contest with Chile. Now that Argentina have qualified for the knockout rounds, head coach Lionel Scaloni may decide to rest Messi for Sunday's game.
Scaloni has confirmed that changes will be made to his starting lineup and those who did not feature in Argentina's first two group matches "will play" against Peru.
A change in goal could therefore be on the cards, with both Franco Armani and Geronimo Rulli set to battle for a starting spot between the sticks ahead of first-choice shot-stopper Emiliano Martinez.
Lucas Martinez Quarta and German Pezzella are yet to feature at this summer's tournament, while Nicolas Otamendi has made just the one substitute outing, and all three centre-backs will compete for a start ahead of Lisandro Martinez and Christian Romero.
Exequiel Palacios and Guido Rodriguez are in contention to start in centre-midfield, with Enzo Fernandez in line to drop out of the first XI and be given a rest having only just recovered from a groin problem.
Should Messi also be rested, Lautaro Martinez could start his first game of the tournament after scoring two goals in two games as a substitute, while Manchester United's Alejandro Garnacho will be pushing to make his Copa America debut at the expense of either Nicolas Gonzalez or Rodrigo de Paul.
Argentina possible starting lineup: Armani; Montiel, Otamendi, Pezzella, Acuna; Mac Allister, Rodriguez, Palacios; Di Maria, La. Martinez, Garnacho