Long-simmering problems in Europe's banking sector are now threatening to boil over in the wake of Brexit and an ensuing slowdown in the region's economy.

Of immediate concern is Italy, whose banks are riddled with bad debt to the tune of €360 billion (S$535.3 billion), or a fifth of the country's gross domestic product.

Now the worry is that one of these banks is close to collapsing. Monte dei Paschi di Siena, the world's oldest bank and Italy's third biggest by assets, has lost 99 per cent of its value since 2008.

It has been bailed out twice since but is still struggling under the weight of bad loans and weakening deposits, as Italians have moved their savings to stronger banks.

Italian Prime Minister Matteo Renzi has asked for an emergency €40 billion taxpayer-funded bank bailout. But this was vetoed by Germany, citing European Union rules that bank bailouts must be funded by bond-holders before taxpayers.

Now some, including Societe Generale chairman Lorenzo Bini Smaghi, are warning that a collapse in Italy's banking sector could spread to the rest of Europe.

With the fate of Italy's banks hanging in the balance and fears of contagion across the continent, investors are rightly nervous.

Shares in the country's biggest banks have plunged by as much as half since April, and shares of banks across Europe have also plummeted this year, with the selldown worsening since Britain voted to leave the EU on June 23.

Still, Bordier & Cie's chief investment officer in Singapore, Mr Bryan Goh, said Europe's biggest banks are still well-capitalised and, if things really go south, it is likely only the smaller, more domestic banks that are at risk of folding.

The crisis could even result in opportunities for Asian banks, he said.

"European banks have withdrawn a lot from corporate lending and trade finance in Asia and they may retreat even more now, which is an opportunity for local banks to fill that space."

Standard Chartered economist Edward Lee was more circumspect.

"A more severe and sharp onset of European banking sector problems may directly affect interbank liquidity and funding (in Asia). Indirect impact through growth and sentiment will follow," he said.

Europe's banks have been troubled since the 2008 financial crisis as European economies have largely stagnated in its wake, making it hard for banks to make profits.

Furthermore, the European Central Bank has kept interest rates in negative territory since 2014. While this is meant to spur lending, it has also made it tough for banks to maintain and grow their bottom lines.

Brexit has only made things worse, with economists predicting that it will lower GDP growth not just in Britain but also across the continent in time to come.

British banks have been warned by ratings agency Standard & Poor's that their earnings will likely be hit as Brexit will hurt consumer confidence and demand for credit.

The International Monetary Fund, for its part, is most concerned about Germany's Deutsche Bank, saying in a report late last month that it is the riskiest bank in the world and a potential source of external shocks to the global financial system. The warning came just after the bank failed a US Federal Reserve stress test for the second year in a row.

It is not news that Germany's biggest bank is facing problems. In October last year, it unveiled a massive restructuring plan that would involve axing some 9,000 jobs and exiting 10 markets, as it took a massive writedown in its investment banking business.

German Finance Minister Wolfgang Schaeuble said earlier this year that he considers Deutsche Bank "rock solid", but investors are not convinced. The bank's share price has plunged 70 per cent this year.

There are more pockets of banking weakness in the rest of Europe.

Greece has injected cash into its banks three times, to almost no effect. One of the largest private banks in Greece, Piraeus Bank, for example, is now worth less than €1.5 billion even though it received a capital injection in December, when it was valued at €4 billion.

What is becoming clear across the continent is that investors want action, and they want it now.

As SocGen's Mr Bini Smaghi told Bloomberg: "What's needed is a European solution. So far, we've had national solutions. We need a clear backstop."

However, UBS Wealth Management's regional chief investment officer for the southern Asia-Pacific, Mr Kelvin Tay, told The Straits Times that real change will take time.

"The most comprehensive tool banks have for improving profitability is cost reduction through digitisation , which would lower expenses and make branch networks leaner. However, this is a multi-year process rather than a short-term fix," he said.

"Additionally, in the first years, it is very likely that banks will face restructuring charges, which will offset the cost benefit."