AS claims that Zinedine Zidane would like to keep the Croatian international in his ranks ahead of the new season but a lucrative offer could tempt Los Blancos to part ways with the player following the signing of Dani Ceballos from Real Betis.
Kovacic joined Real Madrid from Inter Milan in the summer of 2015 following the arrival of Rafa Benitez to the club.
The 23-year-old midfielder has since failed to establish himself as a regular in Los Blancos' line-up due to the huge competition for places in Zidane's 4-3-3 formation.
Luka Modric, Toni Kroos and Casemiro have been unmovable since Zidance took over from Benitez, with Isco also being the preferred option when Real Madrid have played with a 4-4-2 system.
AS claims that the Croatian's situation is expected to be even more complicated in the new campaign after Real Madrid have bolstered the midfield further with the addition of Ceballos from Betis in a deal worth €16.5m (£14.5m, $19m).
Furthermore, Marcos Llorente is also expected to stay at the Santiago Bernabeu after the holding midfielder impressed while on loan at Alaves during the 2016-2017 season.
Kovacic is still reportedly likely to continue at the La Liga giants but AS believes that Real Madrid could still cash in on with his services if a good offer arrives.
The situation should put Tottenham on alert with Spurs having been linked with his services since the last summer transfer window.
AS, nevertheless, claims that Mauricio Pochettino's side already made a €25m offer to lure him to north London earlier in the summer as the Argentine boss is a big fan of Kovacic.
Tottenham may consider to reignite their interest in the player if Real Madrid put him on the market but AS points out that they could face competition from Inter and AC Milan.