The Brazilian-born forward has become one of European football’s most sought-after forwards over the course of the past year, and even came close to joining Liverpool last summer.
His impressive tally of 21 goals in 24 La Liga games this term has led to talk that Chelsea will trigger his release clause – thought to be around £35m-£40m – in six months’ time, but Spurs are also keen.
Tim Sherwood is said to be ready to move for a top class front-man, with speculation suggesting that he has lost patience with Roberto Soldado.
Tottenham are capable of paying out a large sum for Costa, with last summer’s £110m outlay largely offset by the sale of Gareth Bale.
Chelsea are confident of winning the race, but the Liliywhites will not give up hope.
Spurs can also be buoyed by claims that the Blues have made Radamel Falcao of Monaco their prime target, which could leave the door open to snatch Costa.