The tweet, which has since been taken down, said: "I ain't going to say attack don't let the media make u believe that was terrorist that did it. #OTIS."
It is understood that OTIS stands for Only The Illuminati Succeed.
The 20-year-old, who has over 39,000 followers on the microblogging site, has since tweeted: "If you don't like what I have to say un follow me!! Some things get took way out of context..."
However, Liverpool said in a statement: "The club takes this matter extremely seriously and senior club officials have informed Nathan Eccleston that we are undertaking an investigation into the circumstances surrounding these postings and will decide on an appropriate course of action."
This is not the first time that a footballer has got into trouble using Twitter. Former Liverpool player Ryan Babel was handed a charge by the Football Association in January for posting a mocked-up picture of referee Howard Webb wearing a Manchester United shirt after Liverpool's 1-0 defeat to United in the FA Cup.
Then in April, West Ham defender Danny Gabbidon was charged with improper conduct by the FA after he posted an offensive comment on Twitter.
Separately, Twitter filed a legal action on Monday against a company that trademarked the word "tweet" before the microblogging site could do so.