The rate or yield on Spanish 10-year debt hit 5.562 percent from 5.526 percent on Friday at the close, the highest rate since 2000, before dropping to 5.554 percent at midday.
The rate on Portuguese 10-year debt rose to 7.139 percent from the closing value on Friday, when it reached a record of 7.193 percent during trading.
Investors are concerned that when Spain and Portugal issue more debt this week, they will have to offer exceptionally high interest rates to attract lenders.
Much of the focus was on Portugal, following weekend reports that France and Germany were seeking to push it to accept a bailout from the European Union and the International Monetary Fund.
Portuguese Prime Minister Jose Socrates rejected over the weekend a report by German weekly Spiegel that it was being pushed into a rescue, while Paris and Berlin also denied this.
Lisbon argues its situation is different because its deficit and debt are lower, it has no property bubble and its banks are not troubled.
"Portugal will not need any external aid," Spanish Finance Minister Elena Salgado told Cadena Ser radio when asked about the possibility.
"I think that Portugal will not need any bailout because it its meeting its targets. It has structural weaknesses but it is carrying out the needed reforms," she added.
Fund managers will also focus on the extent to which the offers are oversubscribed or undersubscribed.
Usually, debt issued by top-flight countries is oversubscribed, and the rates demanded by lenders are relatively low, because top-flight government debt is considered one of the safest investments for savings funds, banks and insurance companies needing to hold long-term, low-risk debt.
On Wednesday Portugal intends to issue debt for three years and for nine years to raise 750-1.25 billion euros ($968-1.61 billion).
"The real test for Lisbon will be Wednesday: are investors going to put their hands up?" said Cyril Beuzit, a strategist at BNP Paribas.
"There aren't a lot of alternatives, Portugal is going to need to appeal to the European Financial Stability Facility," he said.
On Thursday, Spain is to issue five-year debt to raise 2.5 billion euros.
For several weeks, Spanish and Portuguese debt has been under heavy pressure on bond markets because these countries have big public deficits and weak prospects for growth.
When investors perceive that risk attached to a government's debt has increased, they mark down the price of that debt or sell it, with the effect that the fixed interest for the life of the debt rises as a percentage of the new lower price.
This sets a benchmark of the interest rate which the government concerned must offer to attract lenders when it next issues debt.
There have been signs of tensions recently also on debt issued by Italy and Belgium, two other members of the eurozone. Rates of 6.0% or more may become prohibitive.