The cost of 10-year borrowing for Italy fell below 4.5 percent on Thursday for the first time for nearly two years, and Spanish rates also eased to the levels of March this year.
The yield on 10-year Italian debt on the market for existing bonds fell to 4.496 percent from 4.591 percent at the close of trading on Wednesday.
The last time the rate was below 4.5 percent was in December 2010.
The rate on 10-year Spanish debt fell to 5.222 percent from 5.328 percent.
The yields on debt issued by Italy and Spain had fallen sharply on Wednesday, and BNP Paribas bank bond strategist Patrick Jacq said this was a sign that the eurozone sovereign debt market was returning to "normal" conditions.
This reflected sentiment that the situation of debt-stricken countries on the periphery of the eurozone would improve in coming months, he said.
Investors have been reassured in particular by an agreement on Tuesday between the International Monetary Fund and the eurozone on repackaging a rescue programme for Greece.
The agreement means that, subject to the success of a Greek bond-buyback, Greece will receive the latest instalments of current bailout funds to avert bankruptcy and that its long-term debt mountain will be reduced.
The agreement "pushes a bit further away the risk that the eurozone could explode," Jacq said.
This threat had undermined investor confidence in the eurozone and pushed borrowing rates up sharply for the countries most overburdened by debt.
Jacq said Italy in particular was benefiting from economic reforms put in place by Prime Minister Mario Monti.
Spain was benefiting from improved conditions for it to request help from the eurozone if it got into more severe difficulties over its public debt, and this was pushing its borrowing rates down.
The borrowing rate for France, considered to be among the stronger countries in the eurozone, continued to fall, to 2.024 percent from 2.050 percent.
This reduced the so-called spread, or risk premium, above the 10-year German borrowing rate, the benchmark for the eurozone, to 0.64 percentage points.
Jacq said this indicated that a decision by Moody's rating agency to downgrade France's top-level rating by one notch had not any effect on French debt