The 23-year-old impressed at Euro 2016 and helped Portugal win the tournament, making six starts in the competition, including the 1-0 final win over France.
The Sporting Lisbon midfielder has been linked with a move after impressing in France, with Chelsea believed to be interested, but it now appears Jurgen Klopp tried to land the player, as well as Inter Milan.
According to Portuguese paper Diario de Noticias, Mario was the subject of a €40m bid from the Reds, which was rejected.
“Proposals for the lad don’t come directly to me,” Mario’s father is quoted as saying.
“They go to his agent, Kia Joorabchian, and the Sporting president. I was informed, however, that the club said no to €35m plus €10m in bonuses from Inter and to €40m from Liverpool.
“Nothing particular has happened, my son is a professional and knows he has a contract with Sporting.
“At the same time, we’re talking about a very valuable player for the club, who would aspire to bigger projects in the course of his career.”