The Football Association upheld the decision which means the England striker is now banned for three games.
"There is nowhere to go other than to seek some kind of legal redress," said West Ham co-chairman David Gold.
Rules allow clubs to take disputes to arbitration and if necessary to the Court of Arbitration for Sport.
The FA has refused to comment on the latest development.
Carroll, 25, was sent off after clashing with Swans defender Chico Flores during his side's 2-0 victory.
He will now miss the Premier League games against Aston Villa, Norwich and Southampton.
In apparent reaction to the decision, Carroll, who set up both goals tweeted: "Disgraceful."
According to the FA, West Ham failed to prove to a three-man panel that referee Howard Webb had made "an obvious error" in dismissing Carroll after the forward's arm hit Flores's forehead.
However, West Ham manager Sam Allardyce was certain that Webb made a mistake.
"There has to be somebody to look at that in the cold light of day and say that is certainly not a red card," he said after the win.
Gold added: "We are hugely disappointed at the outcome of the process.
"The last thing I want to do is going to some kind of legal issue because I think it is a footballing issue.
"But we are fighting for our lives. If we were mid-table we would probably get on with it but we are fighting for our lives to retain our Premier League status and we owe it to our fans, we owe it to ourselves.
"We are upset, we feel we have been badly treated.
"Most judgements are not made by three people, they are made by 12 - that's why they are called juries.
"I believe if you had gone to the FA Council and asked the 100 members 75% would have said it was not a sending off, but if you just take three people it may not be representative."
West Ham are 18th in the Premier League, one point from safety.
It is not the first time the club has been involved in a legal battle. Sheffield United took legal action in 2007 when they were releagted and West Ham stayed in the Premier League after it was alleged Carlos Tevez was ineligible to play.
The long-running dispute between the clubs over Tevez was finally settled in 2009 when the clubs reached an out-of-court settlement.
In a joint statement at that time the agreement was described as a "satisfactory settlement for compensation" and the figure, which was not disclosed, is understood to be more than £10 million.
The compensation claim began after United successfully argued that Tevez's participation in crucial matches in the 2006-07 season, while in breach of Premier League third-party rules, played a material role in their eventual relegation.