DeMint, who previously said he would not run, said he has discussed a campaign with his wife and will pray on his decision, The Hill reported Wednesday.
"Out of respect for the people who have asked us to think about this, that's what we're going to do," he said.
Since 1980, every Republican winner of the key early primary in DeMint's home state of South Carolina has taken the nomination.
Two draft-DeMint movements are forming. One is organized by the veteran conservative operative and direct-mail pioneer Richard Viguerie. The other, Conservatives4DeMint, claims 4,700 members in 35 states.
"I've asked him about the presidential thing twice in the last five or six weeks," said Viguerie, who held a conference call Saturday. "I think he's giving it serious consideration. Hopefully this will push him over the line."
Conservatives4DeMint leader Angela Toth said: "I know Senator DeMint has no personal ambition to be president, but I hope he will answer the call of the American people and run."