Manchester United have been heavily linked with a move for the centre-back as Jose Mourinho looks to strengthen his squad in the summer.
According to Spanish outlet Diario Gol, Umtiti has demanded a pay-rise at Barcelona as he finds himself one of the lowest earners in the squad with £2.6million a year.
The defender wants a figure closer to £7m a year and he is apparently willing to force through a move to get his increase.
United are thought to be willing to pay the substantial amount in a bid to close the gap to Premier League leaders Manchester City.
But now Real Madrid president Florentino Perez has set his sights on the centre-back as they plan a mass summer shake-up.
Los Blancos have had a nightmare year trying to defend their La Liga crown as they sit third in the table.
Diario Gol add that Umtiti has a release clause of £53m, which is considered cheap in the current market.
If both clubs make a move for the 24-year-old then it could lead to a war over wages.
United showed they are willing to push the boat out for their key transfer targets as they offered Alexis Sanchez £600,000-a-week to move from Arsenal in January.
If Umititi leaves Barcelona then there has been talk of them making a move for Tottenham centre-back Toby Alderweireld.
However, Spurs will be hoping to keep hold of their best players to launch a Premier League title charge next season.