Spurs are in the middle of a managerial search and the club are looking at several candidates across Europe.
The north London club are carrying out due diligence on several names but Enrique is believed to be a key target.
Tottenham are planning to hold face-to-face talks with the Spaniard in the coming days and weeks.
And they have received a boost in the pursuit of Enrique as Chelsea are cooling their interest in him.
The Blues are also looking for a new permanent manager and the Spaniard has been on their radar as well.
The former Barcelona boss was also in London earlier this month for talks with the west London club.
However, he is unlikely to be invited for another round of talks as he is no longer a leading contender for the job.
Chelsea have held conversations with Julian Nagelsmann and the German is believed to be their favoured candidate.
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