Barkat had intended to announce the building of a new park in the Silwan neighbourhood outside the Old City known as al-Bustan to its mostly Arab residents.
But Netanyahu warned that, while the city should do as it sees fit, going ahead with the project now would serve "interest groups that want to cause disputes and show Israel in a distorted light at home and abroad."
Barkat wants to reshape parts of al-Bustan into parkland and hotel and business areas to boost tourism to the city.
"This plan benefits residents and tourists and will add to the beauty of Jerusalem," he has said.
Media reports have said the residents to be evicted would receive permission to build elsewhere in Silwan.
The area was occupied by Israel in the 1967 Middle East war and annexed in a move not recognised internationally.
Al Jazeera's Jacky Rowland, reporting from Jerusalem, said a hugely effective grass root campaign by the people of Silwan has been able to reach out to the international community via UN agencies and various foreign consulates in Jerusalem.
"It would appear that Prime Minister Netanyahu was afraid that if the mayor went ahead with announcing his plans, this would have an extremely bad impact on Israel's image, an image which is not too shiny at the moment bearing in mind all the criticisms over its suspected involvement in last month's assassination [of an Hamas official] in Dubai.
"[But] people are under no illusions that the mayor will later dust off these plans and have another go at pushing them through."
She said Barkat has "rather grandiose plans" to turn the residential area into a theme park built on the idea that this was where the biblical King David had his capital city.
The United Nations has expressed concern over the park project, saying the move could jeopardise attempts to bring Israel and the Palestinians back to long-stalled peace talks.
"Whatever the intentions behind such a project, Israel needs to understand that demolishing Palestinian homes in east Jerusalem demolishes confidence not only among Palestinians but frankly also internationally," Richard Miron, a spokesman for UN Middle East envoy Robert Serry, told the news agency AFP.
"Israel needs to understand that demolishing Palestinian homes in east Jerusalem demolishes confidence not only among Palestinians but frankly also internationally."
The Jerusalem municipality has issued demolition orders over the past year to tens of Palestinian residents in East Jerusalem, on the pretext that the homes were built without the required zoning and construction permits.
Palestinians say building permission is impossible to obtain.
The United States and other Western countries have called on Israel to cease the demolition of Palestinian homes in East Jerusalem.
Marwan Bishara, Al Jazeera's senior political analyst, said that for Palestinians, Jerusalem symbolises the future of the Palestinian cause.
"Unfortunately for Silwan, since 1991, the municipality as well as the government have been going after it, confiscating lands, demolishing houses.
"The most open and embarrassing secret of the Middle East international affairs is something that everyone whispers: 'The Israelis are taking over East Jerusalem for Jewish purposes'."
Tensions have risen in Jerusalem over the past week since Netanyahu announced he intended to include two holy sites, revered by Jews and Muslims, in the West Bank under a separate Jewish heritage plan.
Palestinians say they are concerned the heritage project could impinge on Muslim freedom of worship.
Netanyahu has said those fears are misplaced and the project was aimed only at
making renovations at holy places in need of maintenance.