Italy’s government has set up a backstop fund to shore up troubled banks, setting the stage for the rescue of troubled Italian lender Banca Monte dei Paschi di Siena SpA.

At a cabinet meeting in the early hours of Friday, the government of Prime Minister Paolo Gentiloni approved the creation of a €20 billion ($20.9 billion) fund to help troubled banks. Shortly afterward, Monte dei Paschi said it will apply to tap this fund for fresh equity to shore up its balance sheet.

Following...