The spokesman said the suspect, identified as Rabi Usman Sayed Ahmad, also known as "Muhammad the Egyptian", was linked to a Tunisian man, now dead, who has been identified by investigators as the ringleader of the suspected Islamists who carried out the attacks.
In Rome, Italy's interior minister on Tuesday said one of three suspects arrested on Monday night by Italian police was preparing further attacks.
"One of the three arrested in Italy, a character of notable ideological and operational importance, is probably one of the principal authors of the Madrid bombings and was preparing other attacks," Interior Minister Giuseppe Pisanu said in a statement.
The three men were part of a group close to al-Qaida, Pisanu said.
The operation which led to the arrest of the three was "of great importance because it blocked a dangerous group of terrorists that was close to al-Qaeda (and) it is likely to have further developments".