Recent remarks by current council president Muhsin Abd al-Hamid caused enough concern in the two countries to make Kuwait demand an official clarification and for Jordan to consider one.
Press reports quoted Abd al-Hamid as saying in Baghdad on Saturday that this was not the right time to discuss territorial issues with other Arab countries, but that Iraq may do so in the future.
He was reportedly responding to a question about "lands that were torn away from Iraq, such as Jordan and Kuwait".
"I was asked during a meeting held by Baghdad municipality council about some territories given by the former regime to Jordan," Abd al-Hamid said in a statement on Sunday.
"My answer was that the Governing Council has never thought about such a thing because we want to have the best relations with our brethren neighbouring countries now and in the future."
Abd al-Hamid said his comments were "distorted", but his alleged remarks made headlines in Kuwait's seven newspapers.
A Kuwaiti official told the Kuwait News Agency his country "followed those remarks with much interest and surprise" and was waiting for the Governing Council to clarify the comments.
Under Saddam Hussein, Iraq invaded Kuwait and occupied it for seven months before a US-led coalition drove his troops out in the 1991 Gulf War.
Although Iraqi-Kuwaiti relations resumed after another American-led force toppled Saddam's regime in April, Kuwaitis are wary that many in Iraq believe this oil-rich state was part of Iraq.
A UN report on Iraq is due to be
In Jordan, a government spokeswoman, Asma Khadir, said Amman would demand clarification if the "comments attributed to the current president of the council" were confirmed.
However, Amman is "convinced that such remarks do not reflect the opinion of the council or Iraqis," Khadir added.
In the mid-1980s Iraq and Jordan reached a border demarcation agreement that gave Jordan 70 square kilometers of desert land.
While Jordan claimed the land as its own, many Iraqis believed Saddam gave it to the late King Husayn for his staunch support for Iraq during the 1980-88 Iraq-Iran war.
Meanwhile, the UN secretary-general has said many issues remain to be addressed in helping to rebuild Iraq, a day before his report on a fact-finding mission there is due to be released.
"I think the team has laid the ground work for further progress but there are a number of important issues and questions to be addressed," Kofi Annan told reporters after a one-hour meeting with Japan's Foreign Minister Yoriko Kawaguchi in Tokyo.
"I will release my report of the team's work in Iraq tomorrow in New York," he added.
After meeting with his adviser and Iraqi mission head al-Akhdar al-Ibrahimi, Annan said last week that elections in Iraq before the 30 June handover of power were impossible.
The visiting UN head also told reporters that "international cooperation will be essential as we move ahead and help the Iraqi people regain their sovereignty and build a peaceful, democratic and stable Iraq".
Annan arrived in Japan on Saturday on a five-day visit for talks with Japanese leaders on ways to reform the world body and help war-torn Iraq with its reconstruction.
He is scheduled to hold talks with Prime Minister Junichiro Koizumi on Monday and deliver an address to the Japanese parliament, the first by a UN chief.