The Sunday Mirror says Portuguese defender Pepe is expected to miss the rest of the season and Real boss Manuel Pellegrini wants Ivanovic, 25, as cover.
The Serbian full-back, who can also play in the centre of defence, is emerging one of Carlo Ancelotti's most important players.
Jose Bosingwa is sidelined for at least the next two months, but Real could pile pressure on Ancelotti by weighing in with a bid of £10million.
Ivanovic will feature for Serbia at the World Cup and manager Radomir Antic reckons the defender would fit right in at the Bernabeu.
"He is very physical and is always composed," said Antic. "He has the ability to play for Real Madrid."