Borini, linked with a move to Sunderland throughout the close-season, lasted just 12 minutes of his side's 1-0 loss at Fenway Park before being substituted.
The Italian fell under a challenge and was grabbing his right shoulder before making way for Emre Can.
Rodgers, whose team conceded a 90th-minute goal to Marco Borriello, played down the seriousness of the injury.
"It's probably not as bad as first thought," the Northern Irishman told a news conference.
"He's had problems in the past with his shoulder, but he's only bruised it.
"He'll miss a few days of training. He won't train now for the remainder of the trip here, but it's only going to be a matter of days before he's back training and playing again."
Liverpool will compete at the International Champions Cup in the United States, facing Olympiacos, Manchester City and Milan.
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