Dzhokhar Tsarnaev, who is in the intensive care unit of Beth Israel Deaconess Medical Center, is responding sporadically, the ABC news network reported on its Twitter feed.
NBC's Pete Williams is also reporting Tsarnaev is writing answers to questions from law enforcement.
Tsarnaev is being treated for a gunshot in the mouth that exited the back of his neck, according Boston Police Commissioner Ed Davis, who was interviewed on CBS' "60 Minutes."
He also suffered a gunshot wound to the leg, Davis said.
Much of investigators' attention has turned to a trip his older brother and fellow bombing suspect, Tamerlan Tsarnaev, took to Russia last year and whether Chechen separatists or Islamist extremists may have influenced or assisted the siblings.
The two brothers emigrated to the United States a decade ago from Dagestan, a predominantly Muslim region in Russia's North Caucasus mountains.
Tamerlan Tsarnaev, 26, died after a Thursday night (local time) gunfight with police on the streets of Watertown, the Boston suburb where authorities finally cornered his younger brother after a massive manhunt that shut down greater Boston on Friday.
Dzhokhar Tsarnaev, a naturalised U.S. citizen, was found spattered with blood and hiding inside a covered boat parked in a Watertown backyard.
He apparently was hit by gunfire in the shootout that left his brother dead, but it was not clear whether he suffered additional wounds in a final hail of bullets that preceded his capture.
U.S. Attorney Carmen Ortiz, the federal prosecutor for the Boston area, was preparing criminal charges, according to Boston Police Commissioner Ed Davis.
It was not clear when charges would be filed. Prosecutors did not plan any news conference or announcements on Sunday.
The suspect was watched by armed guards in the intensive care unit of Beth Israel Deaconess Medical Center, where his brother was pronounced dead early on Friday.
"He is not in a condition to be interrogated at this point in time," Davis said in an afternoon news conference.
Davis said earlier police discovered at least four unexploded devices, including one similar to the two pressure cooker bombs used in the twin blasts Monday that killed three people and wounded more than 170 near the finish line of the Boston Marathon.
"I personally believe they were (planning other attacks)," he told CBS television's "Face the Nation."
The men's parents, who moved back to southern Russia, have said their sons were framed.
Tsarnaev was shot in the throat, U.S. Senator Dan Coats, a member of the intelligence committee, told ABC. A source close to the investigation told Reuters he had tongue damage.
"We don't know if we'll ever be able to question the individual," Boston Mayor Thomas Menino said Sunday, also on ABC's "This Week" program.
Runners in the London Marathon held 30 seconds of silence before starting their race on Sunday, while people from the greater Boston area that had been on virtual lockdown all day Friday remembered the victims in church services.
"We must be people of reconciliation and not revenge," Roman Catholic Cardinal Sean O'Malley told a packed Cathedral of the Holy Cross. "The crimes of the two young men must not be justification for violence against Muslims."
In the neighboring city of Cambridge, police stationed themselves across from a home where various members of the Tsarnaev family had lived, advising gawkers and bystanders to move on.
Patricia McMillan, who lives two doors down, said she last saw Tamerlan Tsarnaev in the neighborhood the Wednesday before the bombing, noting he had shaved off his beard and that he was smoking.
TRIP TO RUSSIA
Tamerlan Tsarnaev traveled to Moscow in January 2012 and spent six months in the region, a law enforcement source said.
That trip, combined with Russian interest in Tamerlan communicated to U.S. authorities and an FBI interview of him in 2011, have raised questions whether danger signals were missed.
It was unclear if he could have had contact with militant Islamist groups in southern Russia's restive Caucasus region.
A group leading an Islamist insurgency against Russia said on Sunday it was not at war with the United States, distancing itself from the Boston bombings.
"We are fighting with Russia, which is responsible not only for the occupation of the Caucasus but for monstrous crimes against Muslims," said a statement from Caucasus Emirate militants operating in Dagestan.
The insurgency is rooted in two separatist wars that Russian troops waged against Chechen separatists following the fall of the Soviet Union.
The family emigrated to the United States about a decade ago. The brothers spent their early years in a small community of Chechens in the central Asian country of Kyrgyzstan, a mainly Muslim nation of 5.5 million. They moved in 2001 to Dagestan, a southern Russian province where their parents now live.
Neighbours said they noticed nothing unusual about Tsarnaev, who this summer helped his father renovate his first floor apartment in Makhachkala, a bustling city in Dagestan.
"They say he was a fanatic. I didn't see that," said Madina Abdulayeva, 45, who runs the small grocery shop across the pot-holed street where he used to come to chat. "We're all Muslim here. We're all part of Islam. We all pray."
