WASHINGTON—The directors of one of Spain’s leading construction companies were delighted to land an appointment with Rafael Ramírez, the president of Petróleos de Venezuela, to talk about their plans to bid on a $1.5 billion electric-power project for the Venezuelan state oil giant.

But when they showed up at the JW Marriott Hotel in Caracas, it was Mr. Ramírez’s cousin, Diego Salazar, who received them in the presidential suite, say two people who attended the 2006 meeting. Mr. Salazar got right to the point, they say: The...