As European bank stocks faced another day of post-Brexit bloodshed on Monday, a small ray of hope emerged for Italian lenders.

Italian Prime Minister Matteo Renzi is reportedly considering a series of measures to prop up the country's weak financial sector, already hurting well before a referendum shocker that's still rattling global financial markets.

On Monday in Milan, Italy's Unicredit (UNCFF)  tumbled 7.4% to EUR1.92 ($2.11), while Intesa Sanpaolo (ISNPY) slid 9.7% to EUR1.57. It's been a rough year for each stock - Unicredit is down 69.20% in the past year while Intesa Sanpaolo has lost more than 50% of its value.

No wonder, then, that they were powerless to resist the post-Brexit gravity dragging down financial stocks across the region.

"What's happening to Italian banks is not surprising, because banks have never been cleaned up after the crisis and so are still perceived as weak," said Silvia Merler, an associate fellow with Bruegel, a Brussels-based think tank, currently on assignment in Washington, D.C.

"What we have seen today is how weak banks remain at risk when an exogenous crisis event hits and how in retrospect it was short-sighted to procrastinate on the financial sector reform," she said.

On Monday, Bloomberg News reported that the Italian government is considering a capital injection of up to EUR40 billion for its lenders, potentially in the form of fresh capital or guarantee pledges though no final decision has yet been made.

Any new public money would come on top of the EUR5 billion bank stabilization fund announced in April. Known as Atlas or Atlante, the fund is to be financed mainly by institutional investors as well as banks including state-controlled Cassa dei Depositi e Prestiti, along with insurers and banking foundations.

But that amount may not go very far, for a sector saddled with about EUR360 billion in bad loans in a country whose economy is stuck in the slow lane - 0.3% GDP growth in the first quarter compared to the 0.5% EU average - and with borrowing rates still at record low.

Merler also questioned whether any further government measures would do any good to calm markets.

"The recent initiatives, like the Atlas fund, are not of much use in this situation, but the idea of a public injection can also be very dangerous," Merler said. "It's true that Italy pledged significantly less public support to the banks in the early crisis years, but it remains a country with a huge public debt and growth has not yet picked up strongly, so a public bank rescue may unsettle international investors even more."