The Trust owns more than 21 per cent of shares in the Swans.
Swansea revealed on Tuesday that they have spoken with potential investors who could buy a minority share interest in the club.
The Swans said in a statement they had spoken to "a number of parties" as they seek financing for planned expansion of the Liberty Stadium and a new training ground.
However, the club also said any deal would have to be right for all parties and they had so far not received any formal offers, stressing discussions were "at an early stage."
In a statement, the Trust said: "The Swansea City Supporters Trust have responded to press speculation that there is a bid about to be placed that would see some of the club end up in American hands.
"The Trust owns more than 21 per cent of the shares in the football club and wish to make it known at the moment that they do not believe that there is a need for the club to change ownership or proceed with the deal.
And Trust chairman Phil Sumbler added: "We are extremely proud of the part that we have played in the progression of the football club since the current arrangement was set up in 2002.
"During that time the club has progressed from the wrong end of League Two to the top half of the Premier League, while winning our first major trophy along the way.
"The way in which we (as a club) operate has been recognised around the world as a role model for clubs to follow, and given the current strong position of the team and the club we do not believe that this is the correct time for the ownership or set-up of the club to change.
"We fully recognise the excellent job that the current directors and shareholders have done, and our desire would be for this set-up to remain and for us to continue our progress in the same way that we have for the last 12 years.
"Our message to the other shareholders has therefore been that we do not wish to relinquish our shareholding in the football club."
The Trust added that should an offer be forthcoming then a consultation would take place with their members.
Source : PA
Source: PA