According to MARCA, Real Madrid have been alerted to comments from Rossi, who has claimed that his Hungary captain, Szoboszlai, dreams of representing Los Blancos.
Rossi is quoted as saying: "[Szoboszlai] always had one dream: to play for Real Madrid. Because of the very close and direct relationship I have with Dominik Szoboszlai since he started playing football as a child, Real Madrid has always been his dream.
"I don't know if he'll be able to take that step, because the decision depends on him and also on his club.
"I absolutely don't rule out the possibility of him staying at Liverpool and renewing his contract there, especially because he's highly valued within the club. He's a very important player for Liverpool, and Liverpool is one of the biggest clubs in the world."
Real Madrid star feuding with Szoboszlai
Should Szoboszlai head to Real Madrid in the near future, he could be forced to end his feud with Arda Guler. The two locked horns while on international duty in March 2025, with Guler's Turkey beating Szoboszlai's Hungary 3-0.
Tensions between the pair escalated when Guler shushed Szoboszlai, who retaliated by commenting '1088' on the Real Madrid playmaker's subsequent Instagram post, referencing the sparing minutes he had played for Los Blancos to that point in 2024-25.
Days later, after Szoboszlai posted a picture on Instagram celebrating his wedding with partner Borka Buzsik, his comments were flooded with Turkish fans posting GIFs and images of Guler.
Could Szoboszlai stay at Liverpool? Update on contract talks
There is no guarantee that Liverpool would sell Szoboszlai, particularly with contract negotiations ongoing. However, the 25-year-old, who has been a rare bright point in a troubling season for the Reds, has claimed his future is not in his own hands.
Speaking recently about talks over a new deal, Szoboszlai said: "Nothing yet, to be honest. Nothing, from now on... it is not in my hands. It works this way. There is no progression but if the right offer comes then let's see.
"Let's see what happens, hopefully everyone will be happy at the end. Of course [I want to stay]. I like the city, I like the club, I like playing with these guys, I love Anfield, I love the fans, let's see what the future holds. It is not on me. I would love to [stay]. We will see."
Scoring return for Szoboszlai after red card
Szoboszlai sat out Liverpool's 1-0 win at Sunderland in the Premier League on Wednesday after he received a one-game ban for the red card picked up towards the end of last Sunday's 2-1 defeat at home to Manchester City. He was drafted back into the starting XI for Saturday's FA Cup fourth-round tie with Brighton and was back among the goals, with Curtis Jones and Mohamed Salah also scoring in a 3-0 victory.