The Croatian international could be allowed to leave the Bernabeu this summer after the La Liga champions signed Dani Ceballos, and he will not be short of offers during the final six weeks of the transfer window.
Manchester United have previously been linked with the former Inter Milan player and according to AS, Spurs have now made a move for the 23-year-old.
It has been claimed that the North Londoners have lodged a bid of £22m as they look to make additions ahead of the upcoming Premier League campaign.
Head coach Mauricio Pochettino is yet to make any signings this summer and with Kyle Walker having already been sold for in excess of £50m, there is growing discontent among their supporters.
Kovacic has made 73 appearances during his two seasons at Real, but just 34 of his outings have come from the starting lineup.