Since Britain's shock vote on June 23 for a "Brexit", attention in the banking sector has mainly focussed on Italy, where non-performing loans are causing concern, bank shares have tumbled and confidence has sunk.
Political tensions in Europe have also deepened, with Rome and Lisbon trying to bend EU rules on helping laggard banks but meeting resistance from economic powerhouse Germany and the executive European Commission.
"It's putting the whole banking system under stress," said Gunnar Hokmark, a lawmaker in the European Parliament, echoing the nervousness expressed by investors who spoke to Reuters.
"It will be serious for countries in a fragile situation," said Hokmark, who helped write rules imposing losses on bondholders and large depositors of failing banks which Portugal and Italy want loosened to allow state help.
Portugal's problems have attracted fewer headlines than Italy's but its predicament is potentially no less painful. Data show Portuguese savings are being spent, unlike in Italy, and private debt is much higher.
A euro zone official who asked not be identified said Portugal's situation was as critical as Italy's but it was unlikely to be treated with leniency because it was smaller and posed no "systemic" threat to Europe's financial stability.
Portugal sees it differently.
"Wherever you look, there is a threat or a risk," said Filipe Garcia, a financial expert and consultant in Portugal. "If Brexit unfolds into a crisis of confidence or a financial crisis, it will be more difficult for Portugal."
Banks in both countries are struggling with bad debts and need more capital. Yet, with public debts of about 130 percent of gross domestic product, neither has much scope to help.
Lisbon's 2-billion-euro (£1.7 billion) bailout of Banif Bank last year blew Portugal off course from EU spending targets, raising the threat of a penalty from Brussels.
The European Commission began a formal disciplinary procedure against Spain and Portugal on Thursday for excessive budget deficits in 2014 and 2015.
Portugal now needs to recapitalise its largest lender, state-owned Caixa Geral de Depositos, where bad loans have blown a hole in its finances which some reports say could be as large as 5 billion euros.
Convincing investors may be impossible after the central bank penalised bondholders in another struggling bank, Novo Banco. Some are now suing to get their money back, while attempts to sell Novo Banco have made little headway.
Millennium, which had shown interest, has been discouraged by the post-Brexit chaos, during which its already flagging shares had fallen by a quarter on the day of the referendum result before staging a mild recovery.
Brexit was also cited by the Portuguese finance minister when he revised down his growth forecast.
The International Monetary Fund warned in a recent report that banks had lent a lot to "low-productivity firms". That is borne out by data showing one-fifth of loans to businesses are at risk of not being repaid -- double the level 2011.
Compounding the problem, ECB figures show that while bank deposits in Italy continue to nudge up beyond 1.6 trillion euros, they have fallen in Portugal since 2012.
Private debt in Portugal is almost twice the size of its economy and significantly higher than Italy, according to Eurostat, the European statistics .
"Brexit can impact the economic outlook for Portugal ... that can have an effect on the banks," said Roger Turro, an analyst with credit rating agency Fitch though he made clear he saw this happening only in the long term.
But alarm is growing because Portugal risks losing access to the European Central Bank's bond-buying programme aimed at reviving economic growth in the euro zone, with only one rating agency still ranking it investment grade.
The Portuguese government, led by socialist prime minister Antonio Costa, has frustrated Germany, the strongest and most influential euro zone country, by rolling back earlier economic reforms.
RACE AGAINST TIME
Lisbon is now considering whether to copy the bad bank model used, albeit with only limited success, in Italy.
Central bank governor Carlos Costa has called for a waiver of the strict European rules that require losses on bondholders for banks bailed out by the state.
He described the banks problem as "systemic", a term used in the euro zone debt crisis to signal a catastrophic spillover to the wider financial system.
The plunge in share prices could become systemic if, say, depositors were to panic and withdraw their cash.
There are already examples of similar phenomena. Brexit, for instance, is making it more expensive for banks to borrow, not only in Britain but also in the neighbouring euro zone.
The value of Additional Tier 1 bonds, the riskiest type of debt a bank can sell, has fallen sharply since Britain's vote for a Brexit.
Market data show that this trend, which makes it more expensive for banks to sell such bonds, is more acute for Italy's Unicredit and Germany's Deutsche Bank than it is for British banks.
With the prospect of further stress ahead, time is running out for Portugal and Italy to strike a deal.
German Finance Minister Wolfgang Schaeuble suggested last week that Portugal may need another international bailout to follow a three-year 78 billion-euro financial rescue package agreed with the EU and IMF in 2011.
A second senior euro zone official played down such a possibility and bailout talk has focussed more on Italy.
"If there is no bailout of Italy, then there could be a run on the banks," said Thomas Mayer, who runs a research institute owned by German fund manager Flossbach von Storch.
"If there is a bailout, on the other hand, Italy's debts reach such a high point that it appears unlikely it could ever pay it back."
(Additional reporting by Balazs Koranyi in Frankfurt, Silvia Aloisi in Milan, Axel Bugge and Sergio Goncalves in Lisbon and Helene Durand in London, Writing by John O'Donnell, Editing by Timothy Heritage)
By John O'Donnell and Andrei Khalip