The interest rate which Spain must pay to borrow money for 10 years rose sharply to more than 5.0 percent in early trading on Monday to the highest level since the beginning of April.
The yield, or interest rate, on 10-year debt bonds rose to 5.065 percent from 4.912 percent late on Friday.
The rate on 10-year Italian date also rose sharply to 4.731 percent from 4.619 percent.
The bond borrowing rate is of critical importance, particularly to countries such as Spain and Italy which have come under strong pressure on the debt market in the last two years over the high level of their public debt.
Interest rates on bonds issued by many emerging countries and by eurozone countries in difficulties have risen in recent weeks on signs that the US Federal Reserve central bank might soon begin winding down its special easy-money measures which have supported the US economy and also US investment abroad.
Last week the Fed sent a clear message that it would indeed begin winding down its policy and this pushed stock markets down and bond yields up further, notably on US treasury bonds.
Yields on bonds issued by eurozone countries under pressure such as Greece, Ireland, Spain and Italy had fallen markedly since the end of last year.
This had sharply reduced tension over the eurozone debt crisis, owing to a commitment by the European Central Bank to support such bonds if necessary, albeit on tough conditions requiring reforms.
Governments finance their public deficits and debt by issuing bonds, or debt instruments, for fixed periods at a fixed interest rate for the life of the bond.
If the perceived risk attached to the bond then changes, for example rises, some investors sell the bonds and depress the price. The fixed interest automatically rises relative to the new lower price.
The new rate signals the interest which the government will have to pay when it next wants to borrow.
It was this process which put Spain and Italy, for example, at risk of having to pay unsustainably high interest rates to finance their public spending and debt repayments.