Both countries consider Norway a suitable location for the dangerous work, as it is considered politically stable and has large amounts of water, which is needed for the task, according to NRK.
The request for Norwegian assistance in the task came last week on the sidelines of the UN General Assembly in New York and has been repeated on several levels, NRK reported.
The Norwegian foreign ministry could not immediately confirm the specific details of the report, but said the Scandinavian nation stood ready to assist.
"I can confirm that Norway is looking into implementing the resolution of the UN Security Council," ministry spokesperson Ragnhild Imerslund said.
"Exactly how it will be done is too early to say. We don’t know what contribution we are going to make, but we are looking into various options."
NRK reports that Norway has until the middle of November to come up with a reply to the request made by US and Russia. According to UN Security Council Resolution 2118, adopted late last month, Syria has until the end of June 2014 to destroy its chemical weapons.
Norway currently does not have the equipment necessary for destroying chemical weapons, however the United States may offer mobile facilities to help in implementing the task, NRK said.
On Thursday, the United Nations said the that chemical experts have made "encouraging initial progress" and hoped to begin disabling equipment involved in Syria’s chemical weapon process next week. It is assumed that there are around 1000 tonnes of nerve agents and chemicals in Syria.