Corriere dello Sport (h/t Sport Witness) noted on Tuesday that Real are in pole position to land the Brazil international and added the two parties are talking and "the desire is there to put money on the table."
Liverpool have until recently been linked as leaders in the race for Alisson, but it appears Los Blancos are almost ready to cement their interest with an offer for the Giallorossi stopper.
It's always been Roma's desire to retain the former Internacional man for at least one more season, but South American Football Manager researcher Paulo Freitas hinted it might be in their interest to capitalise on the buzz:
Los Blancos have for the past two seasons been reported as seeking an upgrade on current starter Keylor Navas, and a disappointing domestic campaign this season may be the push to convince the club to invest in the position.
Real have conceded 33 goals in 29 La Liga outings this season, the joint-worst of any side in the top five and 20 more than leaders Barcelona, highlighting the weaknesses they have in the back line.
Real have a trump card over fellow suitors it seems, too, after Alisson recently responded positively to rumours linking him with a move to the Estadio Santiago Bernabeu.
Speaking to Globo Esporte (h/t Marca), he said:
"Those links massage the ego, it cannot be denied.
"People get satisfaction from having recognition. I feel very happy to have that recognition.
"It means that I am doing my job well, that they are watching me.
"Even more so when one hears about Real Madrid."
Alisson looks set to beat Manchester City stopper Ederson to the No. 1 spot in Brazil's 2018 FIFA World Cup squad, and South American football writer Robbie Blakeley suggested it's well-deserved:
Former Manchester United midfielder Owen Hargreaves acknowledged Loris Karius' recent improvements for Liverpool on BBC 5 live Sport, which could lessen their desire to sign a new 'keeper:
It would make Real's attempts to lure Alisson all the easier if the Reds weren't willing to meet Roma's valuation of the player, whose price tag could increase with a good showing at the World Cup in Russia this summer.
It would be in Real or any potential buyer's best interests to get a deal wrapped up before the competition gets under way on June 14 should they wish to avoid the risk of paying an inflated figure.