Finance Minister Pier Carlo Padoan is expected to address parliament's budget committee later on Friday on the issue.
"We are studying an intervention to reduce and mitigate the burden on savers" who lost money following the rescue of the small banks, Prime Minister Matteo Renzi said on Thursday.
Italy's financial daily, Il Sole 24 Ore, said the government was thinking of creating a €100 million ($109 million) solidarity fund, with contributions from both the state and the banking sector.
The four banks - Banca delle Marche, Banca Popolare dell'Etruria e del Lazio, Cassa di Risparmio di Ferrara and Cassa di Risparmio della Provincia di Chieti - had all been put under special administration over the past two years.
A €3.6 billion ($3.83 billion) rescue plan was launched by the government last month using a newly-formed National Resolution Fund, which is fed by the country's healthy banks.
But there have still been heavy losses for some investors and the government has drawn fire for the suicide of a retired man after he lost €110,000 which he had invested in bonds issued by Banca Etruria.
Some 130,000 shareholders and junior bond holders lost money in the rescue.
Under mounting pressure from consumer associations and political opponents, Renzi said he supported a parliamentary decision to hold an inquiry into "what has happened" in the banking system over the past decade.
After the European Commission criticized the sale of "unsuitable products" to small investors, the Bank of Italy's director general Salvatore Rossi said in an interview on Friday that the bank had called in the past for the sale of subordinated bonds to ordinary savers to be banned.
The problem, he told Corriere della Sera, is that Bank of Italy does not have the "authority" to intervene in the matter.