Building firm Bovis has denied acting illegally or improperly in securing the Scottish Parliament contract.
Bovis won the contract for construction
The firm made the assertion after a formal request for a police probe from Holyrood project critic and Scottish National Party MSP Fergus Ewing.
He said civil servants failed to explain to the Fraser Inquiry why the most expensive tenderer was allowed to cut its price and others were not.
Bovis Lend Lease said it did nothing wrong in bidding for the tender.
Mr Ewing has made a formal complaint about the handling of the tender process to Lothian and Borders Chief Constable Paddy Tomkins.
The police will discuss the issue with the procurator fiscal before deciding whether to mount a full investigation.
Mr Ewing has also demanded to know why Bovis was allowed to cut its price by £500,000 at the end of the bidding process to become project construction manager while others did not get an opportunity.
He insisted there is a "strong case" that the law governing public tendering process was broken.
He said: "None of the civil servants have offered any satisfactory explanation of why Bovis was reinstated."
Lord Fraser concluded that Holyrood project director Barbara Doig had been unable to provide him with any satisfactory reason for Bovis being readmitted to the bidding process.
The Fraser Report stated: "I have not been addressed on the legality of allowing such a post-tender variation and I refrain from making any comment in relation to the legal implications of the position.
Fergus Ewing has made a complaint to police
"It does, however, appear to me, on elementary considerations of fairness as between competing tenderers, that if one tenderer was effectively permitted to change a very material aspect of the financial basis upon which its tender was submitted, that is an opportunity which should have been afforded to the
A spokesman for Bovis Lend Lease told BBC Scotland on Monday that the company had done nothing wrong.
He denied that it had been allowed to reduce its bid by £500,000 while its rivals were not and rejected the suggestion that the shortlisting had broken EU rules on procurement.
Scottish civil service permanent secretary John Elvidge is already considering asking the independent Civil Service Commission to investigate whether any disciplinary proceedings should be started.