Rating agency Standard & Poors (S&P) says Allied Irish Bank and Bank of Ireland have improved their ability to meet the cost of any new emergency.
The rating agency revised upwards its assessment of the liquidity positions of both banks.
AIB, led by David Duffy, and Richie Boucher's Bank of Ireland are regarded as having "adequate" liquidity and "average" ability to finance themselves on the markets, S&P said.
Banks' improving funding position is mainly down to the reduction in lending to customers since the height of the credit bubble, and the ditching of illiquid investments, the report said.
The better assessment for AIB and Bank of Ireland in particular is down to the reduced dependence on emergency funding from the European Central Bank (ECB), the agency said.
Refinancing risks were lessening for Europe's banks as they improve their liquidity in the wake of the financial crisis, but there remains a considerable risk for some, the agency said.
The greatest improvements in bank funding and liquidity were seen in Ireland, Spain and Greece in 2013, it said.
The research looked at the biggest 50 lenders in Western Europe. Of those, some have insufficient liquidity and several will continue to require support from the European Central Bank after long-term funding help runs out at the start of next year, S&P said.
Rebalancing their businesses and so called "deleveraging" by selling off units to pay down debt, have helped restore the finances of the banks, the report said.
But S&P said it expected several euro zone banks will continue to require liquidity support when the ECB's long-term refinancing operations (LTRO), which started in late-2011 and early 2012, mature in January and February 2015.
"We therefore consider the ECB's ongoing commitment to conduct refinancing operations with full allotment at least until the end of 2016 a critical stabilising factor in case market sentiment deteriorates sharply," it said.Banks have been steadily repaying the LTRO, and the ECB is offering about €400bn of cheap cash under a new plan (TLTRO) to encourage lending.
S&P said it considered liquidity and funding as a positive on ratings for Standard Chartered (STAN.L) and the German Cooperative Banking sector, helped by their strong retail deposit bases.
(Additional reporting Reuters)