The prosecutor's office said it suspects that Padron and the secretary of the Tenerife federation "favoured the contraction of business" for their personal benefit.
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Club president David Aguilar then reported his suspicions to police and Spanish football authorities.
Di Pierro was appointed coach after an Italian investor took control of the club earlier this season with suspicions now raised over whether that investment was linked to illegal betting patterns.
Police started the probe early last year after a complaint from Spain's Higher Council of Sport, the government's sports authority.
The probe led the state prosecutor's office to suspect that Villar "could have arranged matches of the Spanish national team with other national teams, thereby gaining in return contracts for services and other business ventures in benefit of his son".
The four men were arrested on charges of improper management, misappropriation of funds, corruption and falsifying documents as part of a probe into the finances of the federations.