A private meeting early this week between former US President Bill Clinton and US Attorney General Loretta Lynch had provoked bipartisan disapproval and concerns about a potential conflict of interests while Hillary Clinton was still under federal investigation.

The 30-minute private meeting happened on Monday night at the Phoenix International Airport in Arizona, and was described later by Lynch as "primarily social."

"Our conversation was a great deal about his grandchildren. It was primarily social and about our travels. He mentioned golf he played in Phoenix," said Lynch at a news conference in Phoenix, Arizona on Tuesday.

Lynch insisted that no discussions were made about the undergoing federal investigation into Mrs. Clinton's exclusive use of a private e-mail setup during her stint as US secretary of state.

However, her assurance had so far failed to abate concerns of Clintons' critics about a potential conflict of interests in the federal investigation into Mrs. Clinton.

"In light of the apparent conflict of interests, I have called repeatedly on Attorney General Lynch to appoint a special counsel to ensure the investigation is as far from politics as possible," said John Cornyn, the No.2 Republican in the US Senate in a statement on Thursday.

"This incident does nothing to instill confidence in the American people that her department can fully and fairly conduct this investigation, and that's why a special counsel is needed now more than ever," said the statement.

Response from Donald Trump, the presumptive Republican presidential nominee for the US 2016 election, was not as measured as Cornyn's.

Calling the impromptu chat between Mr. Clinton and Lynch a "sneak" meeting, Trump said in an interview with local media that the incident exposed the rigged nature of the process.

Even some Democrats had frowned at the private meeting.

"I do agree with you that it doesn't send the right signal," said Senator Chris Coons, Democrat of Delaware in an interview with CNN. "I think she should have steered clear, even of a brief, casual, social meeting with the former president."

White House spokesman Josh Earnest on Thursday defended the impartiality and record of independence of Lynch during her 30-year career in law enforcement, saying that US President Barack Obama and Lynch understood "how important it is" for the investigations to be "free of political interference."