They are desperate for England's former captain to come out of retirement and line up in the Copa Libertadores, South America's version of the Champions League.
Bolivar have exceeded expectations and find themselves set for a semi-final showdown with San Lorenzo in July.
President Marcelo Claure, who is close to Beckham, revealed: "I have spoken to David about the chance of playing in the latter stages of the Copa Libertadores.
"We shared a beer and, as always when we meet, we said, 'Can we do this?'
"We have never closed the door on this. It's a dream. We all have dreams and the door is always open for having David as a great reinforcement at Bolivar."
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