Jim Jordan, a conservative Republican, had peppered Sessions with allegations that Clinton, then-FBI director James Comey and the Democratic party conspired to undermine Trump.
Sessions testified at an open hearing in front of the House Judiciary Committee on Tuesday, and Conyers made a decision to base his first line of questioning on the following Trump tweets.
"If he (Mr Sessions) or his deputy authorises a new investigation of Mrs. Clinton, it would shatter norms established after Watergate that are meant to prevent presidents from using law enforcement agencies against political rivals", The New York Times said today.
In 2010, Secretary of State Clinton approved a deal which saw Rosatum acquire a controlling share of Uranium One.
The department also confirmed that it was considering a special counsel to investigate the foundation of former president Bill Clinton, and its ties to companies involved in a deal that sold U.S. uranium rights to a Russian state company. But there is no evidence that Clinton herself was involved in the United States government's approval of the deal.
What's it going to take to actually get a special counsel? [John] Danforth took over that investigation as special counsel - and Mr. Mueller. "We will use the proper standards, and that's the only thing I can tell you, Mr. Jordan".
He did not, however, rule out that a special counsel could be appointed if the proper standards were met.
"After another heated exchange, Sessions said, "'Looks like' is not enough basis to appoint a special counsel".
The committee members say they're also concerned about what they call "potentially unlawful global dealings" and alleged connections between foreign entities and the Clinton campaign or foundation.
US Congressman Adam Schiff tweeted, "If the AG bends to pressure from President Trump and his allies, and appoints a special counsel to investigate Trump's vanquished rival, it could spell the end of the DOJ as an independent institution". "I am not supposed to be involved with the Federal Bureau of Investigation". The president has repeatedly and publicly attacked the department, the Federal Bureau of Investigation and senior officials at those agencies, complaining that he wishes he could have more control over how they pursue criminal cases.