Burkina Faso, Chad, Mali, Mauritania and Niger -- which make up the G5 -- agreed in March to set up the special counter-terrorism operation of 5,000 troops for the Sahel region.
On Tuesday, France presented a draft resolution that would give a UN mandate to the G5 troops to "use all necessary means" to "combat terrorism, drug trafficking and trafficking in persons."
The United States, however, said the mandate lacked precision and that a council statement instead of a full-fledged resolution would be sufficient support.
"We find the mandate of the force way too broad, lacking precision," said a US official, adding that it would "set a dangerous precedent" by authorizing the use of force for a broad range of activities.
Under the proposed resolution, UN Secretary-General Antonio Guterres would be asked to provide a report to the council on ways to support the Sahel force through logistical and financial means.
The European Union has already agreed to give 50 million euros to the regional force, but the United States and Britain are unwilling to commit UN funds for the operation, diplomats said.
"The real issue is money," said a Security Council diplomat, who asked not to be named.
The United States argued that the council did not authorize through a UN mandate the Chadian-led regional force fighting Boko Haram, although it expressed support for its mission in statements.
France is pushing for a UN mandate in response to a request from the African Union that the Security Council authorize the regional force.
"While we support a G5 Sahel joint force in principle as a potentially important example of African efforts to fight extremism ... Security Council resolutions are not the only -- nor always the most advisable -- means of providing necessary political support," said the US official.
A vote on the draft resolution could take place next week.
France carried out a military intervention in Mali in 2013 to drive out jihadist groups, some of which were linked to Al-Qaeda, which had seized key cities in the country's north.
Although the Islamists have been largely ousted from the north, jihadist groups continue to mount attacks on civilians and UN forces in violence that has engulfed parts of central Mali.