Citing unidentified government sources, the Hong Kong newspaper also said authorities would welcome bids from foreign telecoms firms for licences to provide Internet services in the zone.
China's ruling Communist Party aggressively censors the Internet, routinely deleting online postings and blocking access to websites it deems inappropriate or politically sensitive.
Facebook and Twitter were blocked by Beijing in mid-2009 following deadly riots in the western province of Xinjiang that authorities say were abetted by the social networking sites. The New York Times has been blocked since reporting last year that the family of then-Premier Wen Jiabao had amassed a huge fortune.
The recently approved Shanghai FTZ is slated to be a test bed for convertibility of China's yuan currency and further liberalization of interest rates, as well as reforms of foreign direct investment and taxation, the State Council, or cabinet, has said. The zone will be formally launched on September 29, the Securities Times reported earlier this month.
The idea of unblocking websites in the FTZ was to make foreigners "feel like at home", the South China Morning Post quoted a government source as saying. "If they can't get onto Facebook or read The New York Times, they may naturally wonder how special the free-trade zone is compared with the rest of China," the source said.
A spokesman for Facebook said the company had no comment on the newspaper report. No one at Twitter or the New York Times was immediately available to comment.
For Facebook, the world's largest online social network, with 1.15 billion users, China represents an important new market for growth.
Facebook CEO Mark Zuckerberg, who has studied Mandarin and visited China to meet with local Web entrepreneurs, has said that making Facebook available in China is in keeping with his company's goal of connecting the world.
Earlier this month, Facebook Chief Operating Officer Sheryl Sandberg met with the head of China's State Council Information Office during a visit to Beijing. The pair discussed Facebook's importance as Chinese enterprises continue to expand abroad "and various cooperation matters around that," according to a post on the Council's website.
Shares of Facebook were up more than 4 percent at $49.19 in trading on Tuesday, though some analysts attributed the gains to Citigroup upgrading its rating of Facebook's stock from neutral to buy.
Twitter, which is preparing for an initial public offering, could also benefit by being available in China, the world's largest Internet market by users. Still, many Chinese Web users already use similar services, such as Sina Corp's Weibo.
China's three biggest telecoms companies - China Mobile, China Unicom and China Telecom - have been informed of the decision to allow foreign competition in the FTZ, the sources told the newspaper.
The three state-owned companies had not raised complaints because they knew the decision had been endorsed by Chinese leadership including Premier Li Keqiang, who has backed the Shanghai FTZ, the sources added.