On the day Petraeus announced his resignation, Rep. Peter King, R-Iowa, said on CNN there were answers to questions on the Benghazi attack "that only he has."
"He's an absolutely necessary witness," King said.
Sen. Lindsey Graham, R-S.C., said "it is absolutely essential" for Petraeus to speak to the Senate Intelligence Committee.
"I don't see how in the world you can find out what happened in Benghazi before, during and after the attack if Gen. Petraeus doesn't testify," Graham said on CBS' "Face the Nation" Sunday.
Tea Party Republican Rep. Jason Chaffetz of Utah said Tuesday on CNN that it was possible that members of Congress could compel Petraeus to testify and Darling of the Heritage Foundation confirmed that.
"There is precedent for Congress to issue subpoenas for former administration officials," Darling said.
In the 2007 case of former White House counsel Harriet Miers, and in 1989, of former HUD Secretary Samuel Pierce, Congress issued subpoenas for their testimony.
"We could, but I don't want to take our eye off the ball," Chaffetz said. "We have four dead Americans. We've got people in the hospital tonight. We've got embassies all over the world that need time and attention. And I don't want to let, you know, a salacious story distract from the fact that Gen. Petraeus should put his duty, his honor, his country first and foremost and testify before Congress."
White House Press Secretary Jay Carney Tuesday said the president had no say in whether Petraeus will speak about the Sept. 11 attack in Libya.
"It is up to Congress to make decisions about who is called to testify," Carney said, "but the president is confident that Acting Director Morell is fully informed and capable of representing the CIA in a hearing about the incident in Benghazi."
House Minority Leader Nancy Pelosi told reporters it remained to be seen whether Congress should have been informed about the FBI investigation into Petraeus' extramarital affair earlier, but said she expected it was not of high enough importance to require they be briefed.
"I think there's some answers that we have to have about notification to Congress. I don't have any reason to think that there are any national security issues at stake in what has transpired," Pelosi said at a ceremony welcoming new Democrats to Congress. "I think some dishonorable things were done, and the honorable thing has to be to resign or not to go forward."
But Darling said Pelosi was wrong in saying Petraeus' personal indiscretion did not affect national security.
"Clearly, Congress should have been informed, and there's evidence that some members of Congress were informed before the elections," Darling said. "A scandal like that, which could impact the way that a CIA director operates, should be shared with Congress."
ABC News' Sunlen Miller and Z. Byron Wolf contributed to this report.