The Anfield club are understood to be looking for a new goalkeeper following Loris Karius' poor performance in the Champions League final last month and have been linked with a move for the Brazilian throughout the summer.
Brazilian international Alisson is also wanted by Real Madrid though, as Los Blancos look to try and bolster their own options between the posts ahead of next season, under new boss Julen Lopetegui.
Sky Sports reporter O'Connor revealed during the Sky Sports Transfer Talk podcast, as quoted by Sport Review, that the player may be preferring a move to the Bernabeu over Anfield.
O’Connor said: “The biggest factor for Liverpool is that the player seems more keen on a move to Real Madrid.”
Alisson, also called Alisson Becker, made 37 appearances in Serie A for Roma this season, conceding just 28 goals and keeping a total of 17 clean sheets.
He moved to the Italian capital from Brazilian club Internacional for a fee of £7.2m in 2016.
Alisson is in Brazil's World Cup squad this summer, looking to add to his current tally of 26 senior international caps. He played at Anfield in early June with Brazil during a warm-up friendly against Croatia.
Liverpool manager Jurgen Klopp is looking to significantly strengthen his Reds team this summer, with transfer targets in a number of positions, as the side prepare for a title push next season.
The Anfield club have already signed midfielder Fabinho from AS Monaco this summer, with the Merseyside club looking to get their business done early in the transfer market to avoid inflated prices.
Liverpool had also targeted a move for French midfielder Nabil Fekir but talks broke down spectacularly in the eleventh hour, and his current club Lyon say he will remain their player next season.