Bronislaw Komorowski, the president-elect, has said the cross should not remain in front of a top state building.
Some members of the main opposition party, which is led by Jaroslaw Kaczynski, the late president's twin brother, have said the cross should remain put until a permanent monument is built at the site, a call backed by their conservative Roman Catholic supporters.
Security officials dragged away a few of the chanting, praying protesters on Tuesday, including a woman who tried to tie herself to the cross.
But officials decided not to immediately press ahead with the plan to move the religious symbol.
While church and state are technically separated in staunchly Roman Catholic Poland, the church wields some influence in political life.
"The cross will not be moved to the church today," said Jacek Michalowski, the head of the presidential office.
"The level of aggression is too high ... the cross should not be used for political games."
It was not immediately clear whether officials would make another attempt to move it.
Komorowski and Warsaw city officials want the cross moved to the church and worked out an agreement with church leaders to do so.
Tadeusz Nycz, Warsaw's archbishop, said on Tuesday that a compromise should be sought, which would guarantee that a memorial plaque would be put up in front of the palace.
Kaczynski, his wife Maria, and 94 others, including top military leaders, died in the plane crash in Russia on April 10.
The former president's death was followed by an unusual period of harmony in Poland's often-bitter politics, but that has since frayed.
Komorowski defeated Jaroslaw Kaczynski last month in an election for the vacant presidency and is to be sworn in on Friday.
The crash also inspired an improvement in traditionally tense Polish-Russian relations.
Moscow shared in Poland's sorrow and pledged full co-operation in investigating the crash, which happened as Kaczynski flew to a memorial for 22,000 Polish officers murdered by Soviet secret police in the 1940 Katyn massacres.
Despite this, Polish officials have complained this week that Russian officials are being slow in handing over documents needed for a Polish investigation into the crash, focusing on whether anyone in Poland was to blame.
The probe is being conducted alongside the main Russian investigation, in which a Polish official is an observer, with both sides supposed to share their information.
Jerzy Miller, the Polish interior minister, said on Monday that holes in documentation provided by Russia are preventing the Poles from reaching conclusions.
Donald Tusk, Poland's prime minister, said he would ask Moscow for an explanation of delays in making the documents available.
Officials in Moscow had no comment on Tuesday on the complaints.