Spain's borrowing costs have risen to the highest rate since the launch of the euro in 1999.
The benchmark 10-year bond yield hit 6.81%, as optimism about the weekend's Spanish bank bailout continued to evaporate.
Italy's 10-year bond yield rose to 6.28%, a rate not seen since January, as concerns about its finances rose.
The interest rates are seen as unsustainable in the long run for two countries weighed down by huge debts.
Markets continued to be uneasy about the situation in the eurozone. In other developments on Tuesday:
- Ratings agency Fitch downgraded the creditworthiness of 18 Spanish banks, following its decision on Monday to cut its ratings on the country's two biggest banks, Santander and BBVA
- The European Commission recommended a European banking union, with a single regulator to oversee banks across all 27 EU states; but disagreements over the proposals highlighted the problems of finding a solution to the crisis
- Greece raised about 1.62bn euros in a sale of six-month treasury bills, but had to pay an interest rate of 4.73%, up from 4.69% at a previous sale on 8 May
- Signs emerged of the eurozone crisis having a deeper impact on the airline industry, with BA owner IAG saying the region's economic woes have hit its expansion plans, and UK airport operator BAA reporting a big fall in passengers flying to countries such as Greece
- The head of the IMF, Christine Lagarde, urged European leaders to take "decisive steps" to break free of the crisis; in a speech at the Center for Global Development in Washington she said economic and financial stability was critical to addressing global environmental challenges
Growing investor concern about the Spanish bank bailout did not, however, unnerve European stock markets, which finished slightly higher, despite a late afternoon dip.
But bank shares fluctuated during the day. Barclays initially fell 3.8% before closing just 0.4% lower, while RBS was down about 2% in the morning but ended up 0.3%.
In Spain, shares in the biggest lenders Santander and BBVA fell 1.3% and 0.9% respectively following the Fitch downgrades, but then clawed back the losses.
In Asia, the Nikkei 225 index closed 1% down while the Hang Seng index dropped 0.6%.
After an initial relief rally following the 100bn-euro ($125bn; £80bn) bailout of Spanish banks over the weekend, analysts said the markets had returned to a mood of caution.
"This is a realisation that Spain, while providing money for its banks, is going to add to its debt-to-GDP ratio," said Paul Zemsky, head of asset allocation at ING Investment Management.
"They're borrowing more money, not doing anything about growth," he added.
"Today we're not worried about Spain's banking system falling off a cliff, but other than that, nothing's changed."
Rather than soothe fears in the financial markets, the bailout of Spain's banks revealed new doubts over its impact on the country's overall debt level, where the rescue funds would come from, and whether the money would be enough.
It has still to be decided whether Spain's bailout money will come from existing European Financial Stability Facility, or a new fund, the European Stability Mechanism (ESM).
Under the terms of the ESM, loans will have priority for repayment over private sector investors. This has worried investors buying new Spanish bonds.
'Too little too late'
Worries about the fragility of some eurozone economies were not helped by comments from Austria's finance minister Maria Fekter, who on Monday refused to rule out that Italy may need EU rescue money in the coming months.
Although, on Tuesday, she emphasised that there were no signs that Rome would need bailout aid, Italy's Prime Minister Mario Monti called her remarks "completely inappropriate".
While the eurozone debt crisis has been a cause of concern for some time, it has so far affected relatively smaller economies such as Greece and Portugal.
However, as the crisis spreads to bigger nations such as Spain and Italy, there are fears that growth in the region may slow even further.
"The eurozone crisis needs much more than short-term measures - but all the investors are seeing is political infighting rather than a collective long-term plan," Justin Harper of IG Markets told the BBC.
"The European policy makers are slow to react to the issues. They are doing too little, too late."
Analysts said the latest fears arising from the eurozone had added to worries about a global slowdown, hurting demand for oil.
They said that a fragile recovery in the US coupled with slowing growth in emerging markets such as China and India had hit investor sentiment.
"It is a continued set of bad news from all across the globe, and we have been belted with it for months on end without any let up," said David Lennox, a resources analysts with Fat Prophets in Sydney.
In London, the price of Brent crude fell 1% to $96.90 a barrel.
Mr Lennox added that if global economic uncertainty continues and oil production levels remain the same, prices may fall even further.