Cited by Ansa, Sileoni said the job losses were part of union agreements signed in recent industrial plans.
Some 9,000 of the posts are potentially early retirements, he added.
Italy's major banks including Intesa, UniCredit, Monte dei Paschi di Siena (BMPS), Banco Popolare and UBI have either closed or sold 4,439 branches between 2009 and 2015.
Meanwhile BMPS, the world's oldest bank, came a distant last in EU bank stress tests in late July.
The conclusion came shortly after its board announced a rescue plan for shifting €9.2 billion in bad loans that have weighed down the embattled lender, which has become the eurozone's problem child.
Some analysts fear that the bank's bad debts may trigger a banking crisis in Italy and eurozone turmoil.