The helicopters remains out of bounds to passengers despite the aviation regulator saying they are fit to fly.
The Civil Aviation Authority (CAA) issued a statement nine days ago saying it did not believe the accident was caused by an "airworthiness or technical problem" based on the information available, and sanctioned a return to the air for all Super Pumas, including the L2.
But the Air Accidents Investigation Branch (AAIB), which is leading the probe into the crash, told the Sunday Herald it "can't figure out" why the CAA arrived at that verdict, saying it was too early to rule out a technical problem as a factor.
The AAIB insisted after its initial investigations it was too early to draw conclusions on what caused the Super Puma AS332 L2 to crash.
On Thursday, an AAIB special bulletin again failed to be clear over whether a technical problem was a factor in the accident.
It said: "To date, no evidence of a causal technical failure has been identified, however, detailed examination of the CVFDR (cockpit voice and flight data recorder) and the helicopter wreckage is continuing."
The oil industry's Helicopter Safety Steering Group (HSSG) has not removed a recommendation to voluntarily suspend the use of the 13 Super Puma L2s for passengers. The L2s are currently only being used for non-passenger flights.
Union sources say there are concerns there is still no clear position over whether there is a technical issue.
The HSSG is not expected to make any recommendation about the L2 until late this week at the earliest. The next scheduled HSSG meeting of any kind is expected on Friday.
It has sanctioned lifting the Super Puma passenger-flight ban for all other Super Pumas, including 19 EC225 helicopters, one AS332 L1 and a number of AS355s that serve the North Sea.
Unite union's regional industrial officer John Taylor said: "There should not be a decision on the L2 until there is concrete information that there is no mechanical fault. We haven't got that yet. What the AAIB report says is that it wasn't this and it wasn't that - but it doesn't say what it was."
Aberdeen North MP Frank Doran believes that, following discussions with industry representatives, it was looking "more and more likely" that a technical fault was not a factor, but added: "It is it too early to say [that there is not technical issue]; to be absolutely cast-iron certain.
"There might be a different message from the black box. But they [the AAIB] have control of that, and they should by now have some preliminary ideas ... I would have preferred to see a more definite result [from the AAIB] and a longer time period [of grounding of Super Pumas].
"At the same time, I have to accept that there are no indications yet that there was a technical fault, so it is difficult to find a reason [to continue grounding] other than the reaction of the workforce."
Doran has approached Patrick McLoughlin, the Secretary of State for Transport, to call for a Piper Alpha-style public inquiry in the wake of the Shetland crash.
He said: "Eurocopter have something like 800 Super Pumas of one model or another flying around the world and none of them have a safety record like ours, so that's another reason for having a public inquiry, because there is something fundamentally wrong here.
"It needs to be done soon, but soon in civil service terms tends to be months, rather than weeks."