Thompson, who made 73 appearances for England from 2002-2011, revealed his diagnosis in an interview with The Guardian on Tuesday.
"It's the rugby that's put me through this," the 42-year-old said.
"I finished up with nothing really at the end of it.
"You see us lifting the World Cup and I can see me there jumping around. But I can't remember it."
Several media outlets reported that Thompson and other former players were planning to launch a lawsuit against the sport's governing bodies.
World Rugby and England's Rugby Football Union told Reuters they had not been contacted in relation to any legal action.
"World Rugby takes player safety very seriously and implements injury-prevention strategies based on the latest available knowledge, research and evidence," the sport's world governing body said in a statement. World Rugby is also a signatory to the Zurich Concussion Consensus.
The RFU, which has been at the forefront of the investigation into the potential dangers from concussion in recent years, said as part of its HEADCASE initiative:
"The RFU recognises that concussion is an important player welfare issue in rugby and takes its responsibilities very seriously. The RFU has a comprehensive risk management strategy."
Despite much research and a far better protocol for treating potential concussion during matches, the latest high-profile cases underline the risks that still remain.
"Reports of a possible class action lawsuit on behalf of former rugby players should be a wake-up call for all contact sports," Peter McCabe, chief executive of Headway, a UK-based brain injury campaign group, told Reuters.
"It took similar action in the United States to force the NFL to recognise the seriousness of the issue and accept that it simply had to change in order to better protect the short and long-term health of players."
The NFL agreed in 2013 to pay damages that could amount to more than $1 billion over 65 years to resolve a host of concussion lawsuits from former players.
In football, Nobby Stiles and Jack Charlton, who were both members of England's 1966 World Cup-winning team, died of dementia this year, while Manchester United great Bobby Charlton has also been diagnosed with the condition.
Thompson said his diagnosis had come as no surprise.
"It was so brutal during the week that you'd come home on the Thursday for your day off and I'd just be like: 'I don't think I can play, I feel utterly battered,'" he added.
"They had us for that Six Nations period, and the autumn internationals, and they literally just beasted you until you fell apart.... Would I do it again? No, I wouldn't."
