Confirming the 1997 theft of the papers for the first time, the spokesman said the offer had been made recently to a top Church official, Cardinal Angelo Comastri.
"Naturally, as we are talking about stolen documents, the cardinal refused," the spokesman said.
The Vatican did not divulge any further details about the approach or explain why the theft had not been made public when the documents disappeared in 1997.
The missing items were a letter signed by Michelangelo and a document he is thought to have written.
They formed part of a large archive linked to the construction of St Peter's basilica, which was completed in 1626 after nearly a century of stop-start works.
Michelangelo, who was an architect and engineer as well as a sculptor and painter, was brought in to oversee construction in 1547.
Although he died in 1564, he is considered one of the principal architects of a building that has become synonomous with the Roman Catholic Church.