The 16-year-old forward has emerged as one of the stars of the Under-17 World Cup in India, having already scored three goals in three games, including the winner during Brazil's 2-1 victory over Spain in the opener.
AS reported earlier this week that Barcelona have made an enquiry to lure him away from the Flamengo academy with the Catalans having been closely monitoring his development for a while.
The Spanish publication said that the Catalans had already spoken with Lincoln's agents to take the lead in a potential race for his services.
AS added that Barcelona were yet to launch a formal offer to sign the Brazilian starlet but had already been informed that Flamengo want around €30m (£26.8m, $35.3m) to part ways with the teenager.
But it looks like the Catalans will need to act quickly with Marca now reporting that Real are also considering joining the race.
Furthermore, the other Spanish publications suggest that Los Blancos could have an advantage over Barcelona due to their good relationship with Flamengo officials.
Nevertheless, in the summer Real broke the bank secure the services of Lincoln's teammate Vinicius de Oliveira Junior despite previous reports claiming that he was poised to move to the Nou Camp.
Zinedine Zidane's side paid a stunning €45m fee to win the race for Vinicius and agreed to keep him on loan at Flamengo until the summer of 2019 in order to continue his development in his homeland.
Barcelona will be hoping to win this particular battle after they also tried to sign Dani Ceballos from Real Betis and Theo Hernandez from Atletico Madrid before the Spanish midfielder and the French left-back opted to move to the Santiago Bernabeu.