Charles Owenson, 62, and James Costello, 44, received bribes  from ABC Ltd staff members Kevin Balmer, 52, and Brendan Cantwell, 44.
For cash, the council workers helped award contracts to the Edinburgh firm.
The charges related to the maintenance of Edinburgh Council buildings between 2006 and 2010.
The council workers pleaded guilty on Tuesday at Edinburgh Sheriff Court to breaching the 1889 Public Bodies Corrupt Practices Act.
The men also admitted committing fraud and breaching proceeds of crime charges at a hearing before Sheriff Michael O'Grady.
During the hearing, depute procurator fiscal Keith O'Mahoney told the court the scam was so complicated he would need another day in court to explain how it worked.
However, the contents of a legal document given to the press detailed the nature of the charges against Owenson and Costello, both of Edinburgh, and Balmer and Cantwell, both of Livingston, West Lothian.
The first charge details how between 6 January 2006 and November 10 2010, Balmer and Cantwell corruptly gave £42,521 to Owenson and Costello at various locations in Edinburgh.
The two ABC Ltd employees also gave Owenson and Costello £30,249 in hospitality as an inducement to secure contracts for the repair and maintenance of Edinburgh City Council buildings.
The second charge states Owenson acted corruptly whilst working as an officer at the local authority's Property Care Services Ltd.
It states that he received £28,387 at various locations in Edinburgh between 2006 and 2010 as a reward for the council awarding contracts to ABC Ltd.
The third charge states Costello acted corruptly whilst acting as an officer in the same department and received £14,134 as a reward for the council awarding contracts to ABC Ltd.
Between 18 April 2007 and May 2010, at various locations in Edinburgh, Balmer fraudulently caused Edinburgh City Council to pay ABC Ltd £67,923 for work the company carried out on behalf of the local authority.
Balmer took £22,641 from the fraudulent scheme.
Between 19 May 2007 and 29 December 2011, at various locations in Scotland, Owenson breached Proceeds of Crime legislation by laundering £28,387 by purchasing cars.
Costello carried out the same scam between June 2007 and April 2012 at various locations in Scotland.
Sheriff O'Grady adjourned the case until Friday.
Two Edinburgh City Council employees have pleaded guilty to receiving tens of thousands of pounds worth of bribes from building company workers.