The newly-appointed Anfield boss worked with both players at Swansea City and has identified them as leading summer targets as he looks to make his mark on the Reds squad ahead of the coming season.
Goal.com understands that Liverpool will swiftly turn their attentions to Allen once a deal for Borini is completed.
The midfielder was hugely impressive last season for Swansea as he made 36 appearances in his debut Premier League campaign, earning a place in the Team GB Olympic football squad this summer.
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Rodgers is desperate to land the 22-year-old after missing out on Gylfi Sigurdsson to Tottenham and, amid interest from other clubs, is hopeful of striking a deal to sign the £10m-rated all-action midfielder.
Liverpool are already in advanced talks with Roma over a deal worth around £8 million for Borini, the Italy international who made nine appearances on loan at Swansea from Chelsea in the 2010-11 season.
The 21-year-old was co-owned by Parma and Roma but the Giallorossi took full ownership of his contract last month for £3.95m, removing an obstacle for the Reds as they look to add more firepower to the squad.
Borini has declared in June that he will remain at Roma, where he scored 10 goals last term, adding: "It's not my business that there is talk about whether I go there [to Liverpool] or stay."
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However, a move to Anfield is becoming increasingly likely as Rodgers looks to make three or four signings, stating on Monday that the club hoped to add a "real exciting" player this week.
As revealed by Goal.com, Liverpool are also favourites to sign Chelsea forward Daniel Sturridge while they have also been heavily linked with Real Madrid midfielder Esteban Granero as they step up their activity in the transfer market.
"We hope to complete a deal this week which will be a real exciting one for the club," Rodgers said on Monday.
"We don't have a wheelbarrow load of money, there is not the money flying about people think there is.
"Going forward hopefully we can bring maybe two or three more and then with the group of players that I already have it'll bode really well for us and provide us hopefully with that strength in the group to challenge in every competition."