Congressional lawmakers continued the debate over how to legally handle the Boston bombing suspect on the Sunday talk show circuit, as new details about the accused brothers continued to trickle out.

Tamerlan Tsarnaev may have traveled abroad using an alias in recent years, the chairman of the House Intelligence Committee said Sunday.

“We think he may have traveled on an alias to his home country,” Rep. Mike Rogers, R-Mich., said on NBC’s “Meet the Press” of the older of the two brothers authorities say are responsible for the bombing at the Boston Marathon last week. “When he comes back he has a renewed interest in that radicalization.”

It was the latest wrinkle in the unfolding investigation into how and why Tsarnaev and his younger brother Dzhokhar instigated the most destructive terror event on U.S. soil since the 9/11 attacks in 2001.

Rep. Michael McCaul, R-Texas, the House Homeland Security Committee Chairman, said on CBS’s “Face the Nation” that cooperation with Russian authorities would be crucial in the coming days as U.S. officials piece together the events of last week.

“I’d like to know what intelligence Russia has on this as well,” McCaul said, later adding he thinks “it’s very probable that he possibly could’ve been trained” by terror-related organizations abroad.

The debate continued over how to handle the questioning of Dzhokhar Tsarnaev, who remained in serious condition Sunday in a Boston hospital, apparently unable to speak at this point.

Sen. Lindsey Graham, R-S.C., maintained that the decision by Justice Department officials to not read Tsarnaev his Miranda Rights under the public safety exemption was well-founded, and reiterated his stance that he should be treated as an enemy combatant.

“When the public safety exception expires and it will here soon, this man in my view should be designated as a potential enemy combatant and we should be allowed to question him for intelligence gathering purposes,” Graham said on CNN’s “State of the Union.”

Sen. Charles E. Schumer, D-N.Y., countered that could be contested in a civilian court, while Rogers questioned whether Tsarnaev’s answers to investigators would even be needed for a trial.

“I could make this case without a confession from this guy,” said Rogers, a former FBI agent. “If we let the FBI do this they way they want to do it, that would be the right solution here.”

Meanwhile, Schumer and fellow Democrat Sen. Richard J. Durbin of Illinois said they were committed to moving an immigration overhaul bill, despite calls from some Republican opponents that the legislative process should be slowed because of the events in Boston over the past week.

“We have ample opportunity to amend our bill,” Schumer said. “To not do it or to say do it six months from now is just an excuse.”

Durbin said the bill, which would offer a path to citizenship for about 11 million currently undocumented immigrants, would actually boost national security.

“The worst thing we can do is nothing,” Durbin said. “History tells us that we’re doing the right thing.”