"We don't have consensus yet," James Appathurai, a Nato spokesman, said on Friday. "The discussion will continue tomorrow."
Speaking at a news conference in Turkey, Recep Tayyip Erdogan, Turkey's prime minister, said he took a "negative view" of Rasmussen's candidacy and urged Nato to look for someone else.
"Lets look for new alternatives and find a new name," Erdogan said.
"This has nothing to do with Rasmussen personally. We just don't want Nato to get harmed."
Nato officials said the decision could be postponed until June, diminishing Rasmussen's prospects.
Erdogan told Al Jazeera's Frost Over The World programme, broadcast on Friday, that his country had been urged by Islamic countries to veto Rasmussen's leadership bid.
"We are receiving telephone calls from the Islamic world, telling us: 'By God, this person should not become the secretary general of Nato and we have to take into consideration all these reactions'," he said.
Turkey has a further objection to Rasmussen, accusing him of failing to act on Turkish requests to ban a Denmark-based TV station, ROJ TV, linked to Kurdish fighters who have been fighting for an ethnic homeland in Turkey since 1984.
He has also angered Turkey by opposing its membership in the European Union.
Rasmussen infuriated some Muslims by speaking out in favour of freedom of speech during a row over the publication of cartoons featuring caricatures of the Prophet Mohammed in the Jyllands Posten newspaper in 2006.
Both Rasmussen and Barack Obama, the US president, are due to visit Turkey on Monday.
Other contenders for the post of Nato chief include Jonas Gahr Stoere, the Norwegian foreign minister, and Des Browne, the former British defence secretary.
Radoslaw Sikorski, the Polish foreign minister ruled himself out of the running on Friday.
The backing of all of Nato's 28 member states is needed to get the position.