Peter Murrell, who is also Nicola Sturgeon’s husband, has been accused of giving testimony that is “clearly at odds” with other evidence given to the committee.
The allegation of potential perjury rests on his statement that no other messages relevant to the inquiry existed, something the committee believes to be false.
The harassment complaints committee last week wrote to the Crown Office demanding the release of more text messages held by the prosecutor, invoking the never-before-used section 23 powers of Holyrood committees.
The committee has claimed the messages on WhatsApp and text concern the Scottish Government’s handling of harassment complaints against Mr Salmond and are linked to the messages from Mr Murrell describing ‘putting pressure’ on the police in relation to the complaints.
Jackie Baillie, who is a member of the inquiry and interim leader of Scottish Labour, has written to the Crown Office demanding an investigation into Mr Murrell.
In her letter, she states: “It would therefore appear that Peter Murrell was not truthful with the committee. Given that his evidence was taken under oath, I regard this as a very serious matter and I understand from parliamentary lawyers that committing perjury is considered to be a criminal offence.
"As the Crown Office have all the text and WhatsApp messages secured during the evidence gathering phase of the criminal trial against Alex Salmond, you will be in a position to know whether any more exist than the two already in the public domain.
"If that is the case, and particularly if there are more in which Mr Murrell is involved, I am concerned that his evidence to the committee was not accurate.”
A conviction for perjury can lead to a prison sentence of up to five years or a fine.
Speaking on Monday morning, Ms Baillie said an “immediate investigation” was needed after Nicola Sturgeon contradicted her husband’s evidence.
She said: “Mr Murrell’s testimony is clearly at odds with the facts as they have been presented to the committee.
“Yesterday on the Andrew Marr programme, the First Minister explicitly stated that she believed that the committee had access to all messages from Peter Murrell that she deemed relevant, thereby contradicting Mr Murrell’s statement that there were no other messages.
“It is simply not for the First Minister and her husband to deem what is and what is not relevant to the committee.
“We need an immediate investigation into the possibility that Mr Murrell committed perjury, which is a serious criminal offence.
“The committee has faced obstruction and obfuscation at every turn – this must end. It’s time for justice to be done and for those who have sought to obstruct the committee to be held to account.”
The Holyrood inquiry into the botched handling of harassment complaints is examining how the Scottish Government mishandling the process that led to a £500,000 legal bill after it conceded a judicial review on the basis the process was “tainted by apparent bias”.
Mr Salmond was cleared of sexual offence charges in a trial last year.
The Crown Office and the SNP have been contacted for comment.