Forward Sterling, 19, complained of tiredness before England's Euro 2016 qualifier against Estonia on Sunday.
Hodgson has since questioned Liverpool's fitness regime, which is based on a "two-day recovery" system.
Lineker tweeted: "It's time Roy Hodgson and Brendan Rodgers stopped sniping. It's at their player's expense."
Sterling played 45 minutes of England's 5-0 victory over San Marino on Thursday and, three days later, came on as a 64th-minute substitute in England's 1-0 win in Tallinn.
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The Liverpool forward has been criticised by some fans for complaining of fatigue and requesting to "sit out" England's game in Tallinn, while some players such as ex-Manchester United centre-back Rio Ferdinand have defended him.
Hodgson has said Liverpool's training methods may be part of the reason why Sterling did not feel he would be able to perform at his best against Estonia.
At the Anfield club, players do not take part in a full session until three days after playing, with Rodgers saying players with pace need more time to recover.
Liverpool first-team coach Mike Marsh defended the club's approach.
"With England there has been a lot of discussion recently. We don't really change. Our fitness programmes have been well documented," Marsh  told liverpoolfc.com.
"We try to recover the players as best we can to prepare for the games. "We have a couple of days' recovery after the game and we work with the group that don't start (the previous match) and we build up to the next game and once one game finishes we try to recover as soon as possible."
Relations were already strained between Hodgson and Rodgers after striker Daniel Sturridge, was injured on international duty last month  with the Liverpool manager saying the 25-year-old's thigh injury had been preventable.
Former England captain Gary Lineker has urged Liverpool boss Brendan Rodgers and England manager Roy Hodgson to stop "sniping" at Raheem Sterling's expense.