The 22-year-old signed a five-year deal at Anfield on Friday after completing a £15m move from Swansea City to the Merseyside outfit.
And Lucas believes Allen will be an important player for Liverpool over the coming season.
“He had a training session with us and we can see what a quality player he is so hopefully he can be a key player for us during the season,” Lucas told Liverpool’s website.
“It’s one more player to fight for a place and it’s really good for the club. We need more than 11 players to be successful.
“He had a good season last year and he had a great tournament in the Olympics as well. He has confidence and he has experience so hopefully he will do well for us.”
Rodgers, who replaced Kenny Dalglish in May, will be targeting a top-four finish and subsequent qualification to the Champions League in his first season in charge at Anfield.
And Lucas is confident Liverpool can make a bright start to the Premier League season after gaining competitive match practice in their 4-0 aggregate victory over Gomel in the Europa League third-round qualifier.
“It’s always important to play a competitive game and we had the chance to play two before the Premier League season starts,” he added.
“That will give us a good momentum and confidence to have a good start in the Premier League.
“We still have another week to train but I think it was a positive pre-season for us. We have spent a lot of time working and training with the new manager and we will be ready for the start of the new season.
“Hopefully we can start with a good win.”
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