On his platform Truth Social, the president writes that he got the idea after seeing a picture of the prison, when he was thinking about how to best get to grips with the violence and crime as a result of "sleepy Joe Biden's open border policy".
"Then it happened, I saw a picture of Alcatraz", writes Trump, partly in capital letters.
"And I said 'We will look into renovation and reconstruction of the famous Alcatraz'".
"What a symbol it is, and will be!", writes Trump.
He continues to write that the work to reopen the prison began six months ago and that several companies are on board.
The prison, which is located on an island off San Francisco in California, was closed in 1963 and is run today as a tourist destination. Alcatraz has become something of a popular cultural phenomenon and has been the subject of a number of films.
What do you think?