Both Jackie Baillie and Murdo Fraser have or are planning to write to the Crown Office to investigate the SNP chief executive’s appearances in front of the committee.
Ms Baillie, Scottish Labour’s spokesperson on the inquiry, labelled Mr Murrell’s responses as “obstruction and obfuscation” and “an exercise in spin”.
The interim leader of her party said Mr Murrell’s contempt for the committee was “palpable” and that misleading the committee was “tantamount to attempting to mislead the people of Scotland”.
She said: “Murrell could not explain the nature of his text messages relating to the complaints against Mr Salmond, he could not confirm if he discussed the allegations against Mr Salmond with the First Minister, and he completely failed to refute the allegation that he was present for part of the meeting between the First Minister and Mr [Alex] Salmond in his own home.
“Murrell, despite trying to dance on the head of a pin, effectively conceded that there were other text messages relating to the complaints against Mr Salmond.
“When faced with the charge that he may have misled the committee, Mr Murrell replies glibly and seemingly without understanding the gravity of the offence.
“This obstruction and obfuscation is simply unacceptable and this committee will continue to do all it can to get to the truth.”
It is understood Ms Baillie’s letter demanding an investigation by the Crown Office has been received with a response to be sent in due course.
Murdo Fraser, the Scottish Conservative spokesman on the Salmond inquiry, alleged Mr Murrell gave “false evidence” to the committee under oath.
He said: “He gives the impression that he can say whatever he wants with impunity, but in Scotland such actions must surely have consequences.
“The First Minister lied to Parliament and her husband shares the same casual disregard about telling the truth.
"Mr Murrell seems incapable of giving a straight answer. His dismal and shifty performance was a masterclass in evasion.
“What was particularly craven was the attempt to use the female complainers as human shields to deflect the committee from getting the answers the public deserves.”
Responding after the session, Liberal Democrat member of the inquiry, Alex Cole-Hamilton, said it was “astonishing” Mr Murrell came “so unstuck” during straight-forward questioning.
He said: “That was painful to watch.
"Mr Murrell has once again undermined the First Minister's account of events. She told our committee that she took the meeting with Mr Salmond on 2nd April because she feared he was about to resign the party.
"If that isn't party business and something she ought to have shared with Murrell, I can't think what is. Again it suggests that Nicola Sturgeon knew precisely what Salmond wanted to see her about in direct contradiction to what she told Parliament."
The committee is examining the botched handling of harassment complaints against the former first minister by the Scottish Government, which led to a £500,000 legal bill after the government conceded a judicial review challenge on the grounds of the process being “tainted by apparent bias”.
Mr Salmond was also acquitted of sexual offence charges in a trial last year.