Costa is now a free agent following his release from Atletico Madrid.
La Razon says Spurs are now in the fray and Mourinho is desperate for a reunion with his former striker.
Mourinho brought Costa to Stamford Bridge in the summer of 2014 after he fired Atletico to the La Liga title and Champions League final.
Now Mourinho has asked Daniel Levy to offer Costa a contract in north London, as he looks to bolster his attacking options.
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