Bild am Sonntag newspaper quoted US intelligence sources as saying that National Security Agency chief Keith Alexander had briefed Obama on the operation against Merkel in 2010.
"Obama did not halt the operation but rather let it continue," the newspaper quoted a high-ranking NSA official as saying.
News weekly Der Spiegel reported that leaked NSA documents showed that Merkel's phone had appeared on a list of spying targets since 2002, and was still under surveillance shortly before Obama visited Berlin in June.
As a sense of betrayal spread in many world capitals allegedly targeted by the NSA, the spying row prompted European leaders late last week to demand a new deal with Washington on intelligence gathering that would maintain an essential alliance while keeping the fight against terrorism on track.
Germany will send its own spy chiefs to Washington soon to demand answers.
Meanwhile, several thousand protesters gathered in Washington on Saturday to call for new US legislation to curb the NSA's activities and improve privacy.
Merkel confronted Obama with the snooping allegations in a phone call on Wednesday saying that such spying would be a "breach of trust" between international partners.
The suspicion also prompted Berlin to summon the US ambassador - a highly
unusual move between the close allies.
The Frankfurter Allgemeine Sonntagszeitung reported that Obama had told Merkel during their call that he had been unaware of any spying against her.
It did not cite its sources.
Der Spiegel said he told her that if he had been informed of the operation he would have stopped it at once.
Other media reports said that Obama's National Security Advisor Susan Rice had also told German officials the president knew nothing of the spying.
Merkel's office declined to comment on what Obama told her during their talk.
The White House has said it is not monitoring Merkel's phone calls and will not do so in future, but it has refused to say whether the United States has ever spied on her in the past.
Bild am Sonntag said that Obama wanted to be informed in detail about Merkel, who has played a decisive role in the eurozone debt crisis and is widely seen as Europe's most powerful leader.
As a result, the report said, the NSA stepped up its surveillance of her communications, targeting not only the mobile phone she uses to conduct business for her conservative Christian Democratic Union party but also her encrypted official device.
It said US intelligence specialists were then able to monitor the content of her conversations as well as text messages, which Merkel sends by the dozen each day to key associates.
Bild said only the specially secured land line in her office was out of the reach of US spies.
The intelligence gathered was forwarded straight to the White House, without bypassing the NSA headquarters in Fort Meade, Maryland, according to the report.
Bild and Spiegel described a hive of spy activity on the fourth floor of the US embassy in central Berlin, a stone's throw from the government quarter, from which the United States kept tabs on Merkel and other German officials.
If the spying against Merkel began as early as 2002, it would mean the United States under then president George W. Bush targeted her while she was still the country's chief opposition leader, three years before she became chancellor.
Bild said that Merkel's predecessor Gerhard Schroeder was also in the NSA's sights because of his vocal opposition to the US invasion of Iraq.
Bush was also mistrustful of the Social Democrat because of his close ties to Russian President Vladimir Putin, the report added.
As anger simmered in Berlin over the alleged NSA action, Interior Minister Hans-Peter Friedrich sharpened his tone against Washington.
"Surveillance is a crime and those responsible must be brought to justice," he told Bild am Sonntag.
A poll for the newspaper found that 76 percent of Germans believe Obama should apologise for the alleged spying on Merkel, and 60 percent said the scandal had damaged or badly damaged German-US ties.