According to the Liverpool Echo, the Reds have made a formal bid of around 60 million euros plus a further 10M in performance-related add-ons for the Roma goalkeeper, who is also on Chelsea's agenda, in case Thibaut Courtois moves to Real Madrid.
Jürgen Klopp offered what would be a world record fee for the 25-year-old although it falls short of the Italian club's asking price by some £4.6 million.
Liverpool signed Mohamed Salah from Roma last season but despite the good relationship the clubs enjoy the Italian club would not hesitate to sell Alisson to any other club that matches their valuation.
Jurgen Klopp's side beat Roma in the Champions League semi-finals last term before losing 3-1 to Real Madrid in the decider.
Courtois's time at Chelsea could be up in the Blues land Alisson
A €70m deal would make Alisson the most expensive goalkeeper of all time, smashing the €39m Manchester City paid Benfica for Ederson - Alisson's back-up in the Brazil squad - previous year.
Alisson could be heading to Liverpool after all, with the Roma goalkeeper's agent having fanned the transfer flames with his activity on social media.
German keeper Loris Karius made two blunders in the game that cost the Reds the game.
With fans' anger at Karius reaching breaking point, Klopp has chose to swoop for Alisson, but the BBC claims that their world-record fee for a goalkeeper of £62m might not be accepted. The Liverpool manager has also considered Danny Ward during pre-season but is edging closer to addressing a long-standing issue at Anfield with the bid for Alisson.
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Olivier Giroud speaks glowingly of Kylian Mbappe and Chelsea teammate Eden Hazard before France and Belgium lock horns in their World Cup semifinal.