The Chelsea boss is emerging as the only manager with a recent proven track record who is likely to be available this summer.
Thursday will mark one week without a manager since Zinedine Zidane left Madrid after disagreeing with the board over plans for next season.
Spanish media have claimed Allegri had been offered the job but turned it down. That has seen the Conte option gain ground.
Madrid have host of quick alternatives available to them. Arsene Wenger, Guti, Michel and Michael Laudrup could all be recruited with no need for compensation or lengthy negotiations.