The findings of the 60-day review, ordered by Obama, are expected to be completed by next week, the newspaper said.
"The US will base its decision on a determination that Iran's long-range missile programme has not progressed as rapidly as previously estimated, reducing the threat to the continental US and major European capitals," it added, citing unnamed current and former US officials.
The move is likely to ease US-Russia ties, which had been strained over Moscow's objection to the missile shield.
A Pentagon spokesman said the White House would make an announcement later on Thursday to discuss "a major adjustment and enhancement to our European missile defense system".
Anders Fogh Rasmussen, the Nato secretary general, called the decision a "positive step".
Speaking to Al Jazeera, Todd Kent, professor of political science at Texas A&M University, said: "It's hard to imagine they [the US] would announce they were not going to put the missile defence shield in Poland without getting something in return, as that would appear very weak.
"What they want is Russia to be able to put pressure on Iran to slow down on their nuclear aspirations, to stop the flow of arms from Russia to Iran.
"I think Iran's major ally is Russia and this is a way of putting pressure on them, I think Iran without Russia is very marginalised ... so this could be a major blow."
Kent said the "big challenge" for Barack Obama, the US president, was "how do you reassure these folks [Poland and the Czech Republic] that you've made promises to that you're going to be there for their security".
A senior foreign policy adviser in Poland said it would be a "failure" of US policy in central Europe if Washington abandons its plans.
"If that is confirmed, it would be a failure in long-term thinking in the US administration regarding this part of Europe," said Aleksander Szczyglo, chief of national security at the Polish presidency.
|The United States and the Czech Republic signed a missile defence deal in 2008 [AFP]
Szczyglo said the proposed missile shield had not only a military dimension, but also a "political and strategic" one.
A Russian official welcomed the news.
"If the United States is really intending to refrain from its plans to place missile defence facilities in Poland and the Czech Republic then that is of course good news," the foreign ministry source told Russia's Interfax news agency.
At an army missile defence conference last month, Marine General James Cartwright, who was tasked with dealing with the technical issues of the plan, suggested that Washington may have underestimated how long it would take Iran to develop long-range missiles.
Military officials at the conference discussed possible alternatives to the defence shield plan, including using shorter-range interceptors from other locations closer to Iran.
The proposed programme has been under review since shortly after Obama entered office in January.
The missile defence shield prompted severe objections from Moscow, which said it was a threat to Russia's own nuclear deterrent.
Obama has tried to improve relations with Russia and the expected decision comes just a week before Dmitry Medvedev, the Russian president, is due to arrive in the country.
Medvedev has previously praised Obama for choosing to review the plans, though the US administration has maintained that the European missile defence plans were aimed at countering a threat from Iran and posed no threat to Russia.
The administration of George Bush, Obama's predecessor, had pushed hard for the shield, arguing that Iran was developing long-range missiles alongside its controversial nuclear programme.
But Iran denied that it was seeking to build atomic weapons and said its missiles were only for self-defence.
Six world powers - the United States, Russia, China, Britain, France and Germany - are due to start wide-ranging talks with Iran on October 1.