Aljazeera's correspondent in the West Bank, Walid al-Umari, said Fatah announced on Saturday it had control of 14 councils against Hamas' nine.
Hamas at the same time said it had won control of 13 councils.
Al-Umari reported that an unofficial tally on Friday gave Hamas control of seven councils against 11 for the Fatah movement, according to results from 24 of the 26 councils contested on Thursday.
Official radio said on Friday evening that Fatah took 60% of seats to 23% for Hamas.
Official results will be announced on Sunday, instead of Saturday. No reason for the postponement of the announcement has been given, al-Umari said.
Local press reports indicate that Fatah has secured a majority of seats, he added.
Nevertheless the results are certain to send a message of the Islamic group's strength to Mahmud Abbas, the Fatah man who is expected to succeed Yasir Arafat in a 9 January presidential election and then try to restart peace talks with Israel.
The municipal election is seen as a test of strength between Fatah and Hamas, as well as a dry run for the organisers of next month's election.
Hamas is boycotting the presidential election in keeping with its opposition to the Oslo peace process with Israel, initiated by senior Fatah members.
A Hamas win is likely to send a
message to Mahmud Abbas
Electoral commission officials estimated the election turnout at up to 90%.
"The results assure us that Fatah is still in control and we are witnessing a healthy democratic process," senior Fatah official Jibril al-Rajub said. "We also congratulate Hamas."
The Islamic group said it had done even better than the figures appeared to indicate, and was waiting for the final results.
"[This] indicates that Hamas represents the Palestinian people well, and that the Palestinian people are eager for reforms and an end to an era of corruption," Hamas spokesman Mushir al-Masri said.
Hamas opposes Israel's right to exist on stolen land, advocates armed resistance to Israeli occupation and aims to establish an Islamic state in historic Palestine.
It also calls for reforms in a Palestinian Authority that many see as corrupt and out of touch. Hamas has won favour for charitable work that has helped to replace crumbling official public services.
"The results assure us that Fatah is still in control and we are witnessing a healthy democratic process"
Senior Fatah official
On the other hand, Fatah presidential candidate Mahmud Abbas has urged Palestinian resistance groups to lay down their arms, and wants to negotiate a two-state solution with Israel.
Since Arafat's death, polls have appeared to show strengthening support for Fatah.
One this week gave Fatah a nearly 42% trust rating, up from 26% in June. Hamas had slipped to 20% from 22%.