The expansion of special counsel Robert Mueller's investigation would represent the clearest legal threat to date for Trump, who has long maintained that he is not personally under investigation - and who reportedly pressured former FBI Director James Comey to say so publicly.
"Think he's considering, perhaps terminating the special counsel".
"Whoever leaked [news of the obstruction investigation] was obviously reading that he was thinking about giving Mueller the boot", an official told the Daily Beast.
The president sacked Comey in early May.
He told Woodruff he thinks firing Mueller "would be a very significant mistake, even though I don't think there's a justification. for a special counsel".
Gingrich's comments followed Comey's testimony before the Senate Intelligence Committee on Thursday, in which he told lawmakers that he had instructed a friend to share an unclassified memo of an interaction between him and President Trump with a reporter. He told MSNBC's "Morning Joe" that "I don't think the Congress would sit still and allow the president to pick his own investigator".
Accusations of obstruction arose last month when Mr Trump sacked Federal Bureau of Investigation director James Comey. Gingrich said he is troubled by Democratic donations of Mueller's picks to help lead the probe.
Republicans in Congress are also shaking off the notion that Trump could fire Mueller.
House Speaker Paul Ryan said Tuesday he would like special counsel Robert Mueller to stay the course, as a Trump ally hinted the president wants to remove the ex-FBI head.
He added that he would it was "strange" that Mueller would accept the special counsel position after having a "confidential conversation" with Trump that may have included the Russian Federation probe.
This would evoke memories of the "Saturday Night Massacre" of 1973, when President Richard Nixon sought to dismiss a special prosecutor. "If there were good cause, I would consider it; if there weren't good cause, it wouldn't matter to me what anybody said".
"I never said I spoke to the President", he added.
Christopher Ruddy, chief executive of the conservative news website Newsmax, made his remarks during an interview on PBS NewsHour. Ruddy did not immediately respond to questions seeking clarification. With respect to the subject, only the president or his attorneys are authorized to comment'.
Peter Carr, a spokesman for Mueller, declined to comment on the issues Gingrich and others have raised.