Spanish 10-year government bonds have been trading at yields above 6.8%, coming close to the 7% considered unaffordable.
Mr Rajoy was speaking ahead of the EU summit that begins on Thursday.
"The most urgent subject is the subject of financing," he said. Spain has asked for funding for its banks, but the country has not been bailed out.
Eurozone countries have agreed to lend up to 100bn euros ($125bn; £80bn) to support Spain's banks.
The yield on government bonds trading on the markets is taken as an indication of the interest rates that governments would have to pay to borrow money.
On Tuesday, EU authorities put forward their vision for the future of monetary union, which they hoped would reassure investors that they could safely lend to eurozone countries, and avoid further bailouts being needed.
Mr Rajoy told the Spanish parliament: "There are institutions and also financial entities that cannot access the markets.
"It is happening in Spain, it is happening in Italy and it is happening in other countries," he said.
There was bad news from the Bank of Spain, which said that the country's recession had probably deepened in the second quarter of the year.
It contracted 0.3% in the last three months of 2011 and another 0.3% in the first three months of 2012.
The Bank of Spain said its latest data suggested further contraction in the second quarter "at a more intense pace".
Italy held a bond auction on Wednesday, selling 9bn euros of six-month bonds with an interest rate of 2.957%, up from the 2.104% it paid at the last comparable auction at the end of May.
The big dispute in the eurozone is whether bonds will ever be issued jointly by all its members in order to keep borrowing costs down for those currently facing unaffordable debt.
Tuesday's release from the EU said that could be part of its 10-year plan involving closer fiscal union.
But German chancellor Angela Merkel seemed to dismiss that, reportedly saying: "I don't see total debt liability as long as I live," having already said that the idea of eurobonds was "economically wrong and counterproductive".