Swansea's Supporters' Trust wants any change of ownership at the club to be completed before the end of the season.
An American consortium led by Jason Levien and Steve Kaplan is in talks about buying around 60% in the Swans.
The trust will retain its full 21.1% stake and says its views "should not be ignored" as negotiations continue.
Having initially been sceptical of the American investment, the trust says in a statement: "Now it is time to move on."
The statement continues: "Club leadership must be united and the future ownership structure of the club determined before the end of the season to enable the club to be as effective as possible in the transfer market over the summer.
|Business Management lecturer Daniel Plumley from Sheffield Hallam University on Swansea takeover|
|"It's probably a little bit light in terms of valuation.|
|"You might argue that a club in Swansea City's position might be worth maybe in the region of £50m more in terms of a valuation.|
|"Because the bottom club next season will receive £100m roughly from broadcasting revenue alone."|
"We talked last week of our disappointment as an organisation at the situation that had arisen. But now it is time to move on.
"The Trust is looking to work constructively with all the various parties in order to achieve a solution that is best for the football club.
"At the fans' forum last Thursday night, we talked about how a protracted period of uncertainty in the boardroom is not in the best interests of the club."
Levien, the managing general partner of Major League Soccer side DC United, has been in Wales to continue discussions.
Under the terms of the deal, it is expected chairman Huw Jenkins and vice-chairman Leigh Dineen will hold management roles at the Liberty Stadium.
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The trust was disappointed it had been "kept at arm's length from negotiations" after some of its members met Levien on 9 April as news of the potential takeover emerged.
But the trust, which has a representative on the Swansea board, has since studied the takeover proposals and the transaction is to be completed before the Premier League's new record £5.136bn television rights deal starts at the beginning of the 2016-17 season.
"We have to work with the other shareholders, and the other shareholders have to work with us, to reach an agreement that works for all parties and, most importantly, the football club," the trust added in its statement.
"It is our job, as custodians of a significant stake in the club, to ensure we protect the interests of our members whilst moving discussions towards a conclusion.
"Our football club, uniquely in the Premier League, provides a strong 'voice in the boardroom' on behalf of all supporters, and we should not give it up lightly.
"As the second largest shareholder, our views should not be ignored."