The 21-year-old scored his first goal for the Reds in a 3-0 victory over Gomel in the Europa League, while the Italy international’s presence forced Hearts defender Andy Webster to turn the ball into his own net in Liverpool’s 1-0 win in Edinburgh on Thursday.
Rodgers admitted Borini, who was signed for £10m from Roma in July, will have to cope with the expectations of everyone at Liverpool, but the former Swansea boss is confident the forward has the necessary temperament to do so.
“Expectancy is what comes from being at a big club so you have to deal with it both as players and manager,” said Rodgers.
“He is not alone. When you come to a big club there is always big expectation and pressure.
“This is a kid who works well and works hard and he will come good there is no question about that.”
He added: “The most important thing is he is fighting for the shirt.
“The biggest thing in football you need is perseverance. All these players are persevering and want to work.”
Meanwhile, Rodgers dismissed the idea that signing a prolific finisher before the transfer window closes is important for the Reds, insisting his squad has a collective responsibility to improve the club’s strike-rate this season.
“It is no good having a goalscorer scoring 30 goals and no one else scores – I’ve always been against that theory,” he added.
“You are going to need upwards of 70-odd goals to be up around the Champions League [places].
“For me it is a team ethic. Luis Suarez will score goals, Steven Gerrard can score goals, young Borini will score goals.
“But it is a team responsibility, not just one player. You bring in one striker who can score you goals, which is great, but it’s not good for the team if the rest don’t score.”
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