It means former employees at Transport Initiatives Edinburgh (TIE) may be unable to speak about their time on the disastrous project.
SNP MSP Marco Biagi said it was “essential” for the silence orders to be lifted in time for a public inquiry into the fiasco.
TIE, a company set up by Edinburgh City Council, was tasked with delivering a £545 million tram line. However, cost overruns, delays and mismanagement by TIE derailed the high-profile scheme.
Part of the deal agreed by councillors to save the project included axing TIE and handing over control of the project to the council and a consultancy firm.
Most of TIE’s workforce, which stood at around 70 earlier this year, have since taken voluntary redundancy or early retirement.
The total bill for the exit payments is thought to stand at around £2.1m, with employees getting a month’s salary for every year worked.
Taxpayers are also having to find another £5m to plug TIE’s pension black hole.
The Sunday Herald can reveal that 41 staff signed confidentiality agreements as part of their exit terms. The deals are usually insisted upon by companies if staff possess commercially sensitive information.
However, in a TIE context, gagging orders could prevent staff from talking about the problems working on the publicly-funded scheme.
First Minister Alex Salmond has promised an inquiry into the project, a forum that would require openness and transparency.
SNP MSP for Edinburgh Central Marco Biagi said: “I would hope that no-one was pressured or given any inducement to sign a confidentiality clause. When the time is right for a public inquiry and more progress has been made on the project it is essential those clauses are lifted to ensure no information is held back from the people of Edinburgh.”
A council spokesman said: “Given the particular commercial sensitivities surrounding this project, it should be expected that, on leaving, staff are required not to share confidential or other privileged information with anyone.”