Annan's warning came in letters sent to US President George Bush, British Prime Minister Tony Blair and interim Iraqi Prime Minister Iyad Allawi late last week, UN officials said on Friday.
A UN spokesman said the world body would have no comment on the letters.
"We have nothing to say about private communications between the secretary-general and heads of state," the spokesman said.
But speaking in Brussels on Friday, Allawi said time was running out for a peaceful solution in Falluja.
"The window is closing for a peaceful settlement," he told a news conference at a European Union summit in the Belgian capital as US troops surrounded the town, west of Baghdad.
US forces have been bombarding
Falluja for weeks without a break
Witnesses said the US military blocked roads around Falluja on Friday and mounted air strikes on parts of the city ahead of a major offensive to crush the fighters.
US troops urged civilians to flee Falluja ahead of the offensive.
US forces sealed all roads to the city and used loudspeakers and leaflets to tell residents they would detain any man under 45 trying to enter or leave the city.
Marine Colonel Michael Shupp said near Falluja that final preparations were on and that it would be over soon. "We are just awaiting orders from Prime Minister Allawi," he said.
Many residents have already fled the town with a population of 300,000, about 50km west of Baghdad.
"The window is
closing for a peaceful settlement"
Interim Iraqi PM Iyad Allawi
US warplanes and artillery bombarded Falluja on Friday, residents said, and fighters fired mortars and rocket-propelled grenades at US forces on the city outskirts.
There was no immediate word on casualties. Ambulances were waiting for fighting to ease before venturing on to the streets.
The US military said air raids on Thursday night had destroyed a command post, arms caches and resistance positions.
Hospital workers in Falluja said the strikes had killed three people, wounded four and destroyed five houses.
The military also said two marines were killed and four wounded in combat on Thursday in Anbar province, home to Falluja and the city of Ramadi, also expected to be a target of the US assault.