The government of the Castilla-La Mancha region where the dump is located, dozens of kilometres south of the Spanish capital, said it had activated an emergency action plan as it believes the fire may last for days.
Firefighters and helicopters were working to extinguish the blaze, which produced a "toxic cloud... That could affect part of the (nearby) town of Sesena," with its 20,000 residents, the regional government added in a statement.
By late morning, three-quarters of the site had burnt, the Spanish capital's emergency services said on their Twitter account.
Authorities have urged residents nearby to close their doors and windows, and to try and stay away from the cloud of smoke, but no evacuations have yet been deemed necessary.
"No problem has been detected in any (air quality) measuring station," the emergency services said in a bid to ease health concerns, adding there were no injuries.
"Everything points to the fact that this disaster was deliberate," Sesena Mayor Carlos Velazquez told Spanish radio, pointing out that the area had been rained on for several days, which makes an accidental ignition unlikely.
But over the years these started to accumulate, resulting in three-metre (10-feet) high piles.
Environmentalists have for years warned that the dump poses a health hazard, and the town of Sesena has lived in fear of the rubber heap catching fire.
These types of blazes are notoriously difficult to put out and have been known to go on for months and even years, as tyres often continue to burn inside even if they are extinguished from the outside, and easily reignite.
The black cloud emanating from the blaze appeared to be moving south over the day, sparing Madrid and its international airport, according to Vicente Garcia, a spokesman for environmental group Ecologists in Action.
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