"Everything is on track and we are moving forward with the location," El-Gamal, chief executive of Soho Properties, told the Reuters news agency.
Haaretz had reported that leaders agreed to abandon the site to prevent an escalation of anti-Muslim sentiment.
The project, planned near what has become known as Ground Zero, has emerged as an emotional issue, with opponents saying the location of the centre is insensitive to the memory of the nearly 3,000 people who died in the September 11 attacks. Some of the victims' relatives, however, are in favour.
Obama strongly backed the proposal for the centre, sparking fierce debate just months before US congressional elections in which Republicans are trying to take back control of congress from the Democrats.
"As the president said on Friday night, he respects that Americans of all political persuasions will have different opinions on this issue,'' Bill Burton, a White House spokesman, said on Monday, in response to Reid's criticism.
"That is a strength in our country and in the Democratic Party. Senator Reid is a fiercely independent individual and the president believes that is one of his strengths as a leader.''
Obama gave his support to the mosque during an annual White House dinner marking the Muslim fasting month of Ramadan, weighing in on the controversy for the first time.
"That includes the right to build a place of worship and a community centre on private property in lower Manhattan, in accordance with local laws and ordinances.
"This is America, and our commitment to religious freedom must be unshakable," Obama added.
Obama has tried to reach out to the global Muslim community since taking office, and the more than 100 guests at Friday's dinner included ambassadors and officials from numerous Muslim nations, including Saudi Arabia and Indonesia.
Al Jazeera's Rosiland Jordan reports on the debate surrounding the proposed centre
"Our capacity to show not merely tolerance, but respect towards those who are different from us, and that way of life, that quintessentially American creed, stands in stark contrast to the nihilism of those who attacked us on that September morning, and who continue to plot against us today," he said.
New York City Mayor Michael Bloomberg, an independent who has been a strong supporter of the mosque project, welcomed Obama's words as a "clarion defence of the freedom of religion".
But top Republicans including Sarah Palin, the former vice-presidential candidate, and Newt Gingrich, the former speaker of the House of Representatives, have already announced their opposition.
The Cordoba Initiative, the group behind the project, describes it as a Muslim-themed community centre with a view of making it a hub for interfaith interaction, as well as a place for Muslims to bridge some of their faith's own schisms.