They voiced concerns amid claims that two senior figures in the Jimmy Reid Foundation are at loggerheads about the body's direction and purpose.
Reid gained international recognition as one of the trade union leaders during the Upper Clyde Shipbuilders "work-in" in the early 1970s.
He died in 2010, with figures including Sir Alex Ferguson, comedian Billy Connolly, First Minister Alex Salmond and former prime minister Gordon Brown attending the funeral.
The Foundation was set up as a non-partisan think-tank, and has Robin McAlpine as its director and Scottish Labour stalwart Bob Thomson as convener. It has no independent legal status, but is under the umbrella of Left Review Scotland Ltd.
The Foundation's flagship project is Common Weal, which under McAlpine's direction has published a series of left-wing policy papers on subjects ranging from the economy to education.
However, McAlpine has also put forward options for the future of the Common Weal, such as influencing future governments and even forming a political party, although he is said to be sceptical of the latter idea.
He has also suggested setting up a Common Weal digital newspaper, film unit and television service.
An online magazine, edited by Kenneth Roy, last week reported tensions between McAlpine and Thomson on the Foundation's future, particularly relating to the notion of fielding political candidates.
Eileen Reid, one of Reid's daughters, has released a statement to this newspaper on behalf of the late trade unionist's family.
It is understood the Reids sympathise more with Thomson's line of thinking.
The statement noted: "Sadly, Kenneth Roy's piece in the Scottish Review about Jimmy Reid's legacy is a broadly accurate description of the unfortunate state of affairs within the Jimmy Reid Foundation.
"The Reid family's concern is to try as much as we can to protect Jimmy's memory and legacy, and to ensure as far as possible that all those who act in his name do so in a manner which respects in a dignified manner his values and humanitarianism.
"However, there is very little the family can do to resolve the current situation, as we do not have a seat on the Scottish Left Review editorial board which governs the Jimmy Reid Foundation Project Board.
"Therefore although we are concerned, we have no formal power or influence to do anything about the current tensions and disagreements but we are hopeful that these will be resolved as soon as possible by the Jimmy Reid Foundation and the Scottish Left Review."
Thomson, in his own statement, said: "At no time have we discussed far less agreed to stand candidates anywhere.
"Standing candidates would require us registering as a political party and would be contrary to our objectives."
He added: "There have also been reports in the media that the Common Weal was proposing setting up a digital newspaper, film unit, television service etc.
"Laudable as these may be, they would require substantial funding and resources.
"More importantly there have been no discussions on any of these ideas and most of them could be construed as outwith our objectives."
It is believed that the Foundation and the Common Weal may be heading for a formal split.
A Common Weal insider dismissed the statements as "narcissism" and "two people versus a movement".
McAlpine said yesterday: "Common Weal will continue."