The court ruled last year that the building in Givat Zeev, northwest of Jerusalem, must be pulled down by November 5, 2015 and around 200 young men set up barricades around it early Wednesday in a bid to stop the order being carried out, an AFP photographer at the scene said.
Police on Tuesday took the rare step of asking for a delay in demolition, saying that deploying security forces to enforce the operation would be a drain on manpower at a time of almost daily Palestinian attacks and clashes.
A motion filed on behalf of the police by the state attorney's office cited potentially "significant consequences on the security of Jerusalem", as well as the danger of reprisal attacks by Jews.
A court statement said it was extending the deadline for 13 days, while rejecting a request for a three-week postponement.
"The judges... are not making light of the professional assessments of the police and concerns about civil disturbances on the part of extremists," it said.
Signup to our newsletter and follow us on Facebook and Twitter!
"At the same time the grounds given do not justify further postponement of execution of the judgment," it said. "Demolition will be carried out no later than 17/11/2015."
Graffiti was found Wednesday on the walls of the court building in Jerusalem, reading, "Don't destroy a synagogue," and "We want a Jewish state".
The slogans were roundly condemned by Israeli politicians, including Justice Minister Ayelet Shaked of the far-right Jewish Home party.
"Crossing moral red lines and vandalism are unacceptable," she said in a statement. "Debasing state institutions will not advance any goal and should be condemned."
Israeli legal rights group Yesh Din, which represents the Palestinian landowners, said that religious sensitivity should not be exploited to legitimise wrongdoing.
"While the demolition of a synagogue is indeed regrettable, a place of worship should not become a symbol of land theft," it said in a statement.
Advocates of the synagogue accuse the court of being eager to demolish a structure used by settlers while stalling demolition of Palestinian militants' homes.