“It is untenable that this important office is repeatedly brought into disrepute,” LSSA chairmen Ettienne Barnard and Max Boqwana said in a joint statement.
Nxasana, of the KwaZulu-Natal division of the High Court, was appointed by President Jacob Zuma in October 2013.
It was reported last week that former justice minister Jeff Radebe instructed Nxasana to resign just a few days before Zuma announced his new Cabinet.
Radebe allegedly claimed that Nxasana had not been given a security clearance because of past brushes with the law.
This included being tried for murder about 30 years ago. He was acquitted.
Nxasana has refused to resign and has vowed to challenge the matter in court or an inquiry should he be fired.
Justice Minister Michael Masutha was expected to meet Zuma to discuss the matter.
Ministerial spokesman Lawrence Ngoveni said it would be Zuma’s decision on how to handle the matter involving Nxasana.
The LSSA, in common with its constituent members, the Black Lawyers Association and the National Association of Democratic Lawyers, believed it was unconstitutional to hold the murder charge against Nxasana as he had been acquitted.
“Mr Nxasana was found to be a fit and proper person to be admitted as an attorney by the high court. This should extend to his suitability as the head of the prosecution services.”
“Mr Nxasana should not be subjected to protracted public speculation and the office of the national director of public prosecutions cannot be subjected to the instability that has plagued it since the dismissal of Vusi Pikoli in 2009,” Barnard and Boqwana said.
Former National Prosecuting Authority head Pikoli was axed as the country’s prosecutions boss following his decisions to institute corruption charges against disgraced former police chief Jackie Selebi and Zuma.
On Wednesday, Democratic Alliance MP Glynnis Breytenbach, a former senior prosecutor at the NPA, said the party believed Nxasana was being targeted because he permitted the reinstatement of fraud and corruption charges against suspended police crime intelligence boss Richard Mdluli.
The DA was concerned that allegations that Nxasana was asked to resign because of previous run-ins with the law involved “political meddling” in the work of the NPA.