Band member Louis Tomlinson, 20, decided to support the charity after his girlfriend Eleanor Calder learned that her grandfather had been diagnosed with dementia.
she said: "I'm so pleased the boys have decided to support the Alzheimer's Society in this way. My family and I have had first hand experience of how the condition affects every aspect of a person's life."
Fans can bid online for the t-shirt until the auction ends on Thursday. It has been hand-signed by all five One Direction stars.
Around 800,000 people in the UK are currently living with dementia. According to Alzheimer's Society research this is expected to rise to 1.7 million people by 2051.
The money raised from the sale of the t-shirt will go towards improving the lives of those living with dementia and Alzheimer's disease, and fund scientific research to find a cure.
Jenny Hughes, chief executive of the Alzheimer's Society, said: "The boys' support will help us reach new audiences and raise awareness of dementia amongst younger people. Everyone has a role to play in improving the lives of people with dementia."