Luis Suarez (Getty Images)
One of Valdez's main rivals for the presidency, Arturo Del Campo, told local media that he'd heard the recordings five days ago, describing them as "slightly compromising Wilmar's image".
Valdez, though, denied they had any link to his resignation.
"This decision has been motivated solely by family and personal reasons that have nothing to do with the context of the current election," he said in a statement published by the AUF.
"I want to stress that I was under no undue pressure, threats or extortion in making this decision."
Valdez's resignation leaves just two candidates for Tuesday's presidential election: farming businessman Del Campo and Eduardo Abulafia, a businessman that local media claim has close ties to controversial players' agent Francisco Casal.
A former player with Atletico Madrid in Spain and Brazil's Gremio, Casal's company represents the majority of Uruguayan footballers and also owns the television rights to the Uruguayan league in the country.