The former Chelsea captain is being linked with a return to England this month after losing his first-choice status this season under Atletico coach Diego Simeone.
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Gallagher has been linked with Manchester United and Aston Villa in recent weeks - and now Spurs are jumping into the fray.
Football.London says Spurs are intensifying their interest, with Atletico prepared to sell.
A loan arrangement, even with an obligatory permanent option, has been ruled out by Atletico, with the Spanish giants insisting on a straight sale.