The sculpted juxtaposition of two of the world's most potent symbols, Christianity and communism, was made by Jesuit priest Luis Espinal.
A poet and artist, Espinal was assassinated in 1980 by right-wing paramilitaries, and the pope paid tribute to him on Wednesday on his way from the airport to La Paz, stopping at the site of his death.
Pope Francis is the first Jesuit pope.
Local Catholic leaders said the unusual gift fits with Morales' quirky personality.
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"This is a provocation, a joke" said Bolivian Bishop Gonzalo del Castillo.
Bolivia's communications minister, Marianela Paco, explained that the gift, presented yesterday, was symbolic: "The sickle evokes the peasant, the hammer the carpenter, representing humble workers, God's people."
"That was the intention of this gift, there was no other," she told Radio Patria Nueva.
The Vatican said the pope made no specific comment on the gift.
Religious expert Francisco Zaratti said the gift was anachronistic.
"In the '70s such an object made sense, it represented a commitment to socialism. Today after the fall of the Berlin Wall, it belongs to the past," he said.