Archbishop Claudio Maria Celli, who has been tipped as a possible secretary of state, or "Vatican prime minister", said Francis would weigh up appeals made by two cardinals on the eve of his election for the bank - formally known as the Institute for Works of Religion (IOR) - to be scrapped.
He said the Pope would be guided by a desire for "transparency . . . and faithfulness to international laws or rules in this field", including those on money-laundering.
"The Pope will consider some suggestions because during the general congregations (the formal meetings that preceded the conclave), some cardinals were intervening about the problems of the IOR," Archbishop Celli said.
The bank has been surrounded by controversy since the death of "God's banker" Roberto Calvi, the Italian financier found hanged under Blackfriars Bridge in London in 1982.
Italian prosecutors are investigating allegations of money-laundering and a claim that one account was indirectly linked to the Sicilian mafia godfather Matteo Messina Denaro.