So far no party has said it will put up a candidate against Mr Davis - the Lib Dems say they support him on terror detention and Labour is still deciding whether to take part in what some of its MPs have described as a "stunt".
The BNP, which came fourth in the seat in the 2005 general election, says it will not stand against Mr Davis as it agrees with his stance on terror detention.
UKIP, which was fifth, said it was still considering its position, although one of its MEPs, Godfrey Bloom, has offered to campaign for Mr Davis.
If no other candidate has come forward by the close of nominations, Mr Davis would be returned as the MP without a vote.
He has led the opposition to Labour's plans to extend the maximum limit terror suspects can be held beyond the current 28-day maximum.
On Wednesday, Mr Davis accused the government of "buying" the nine votes they needed to get the legislation through the Commons.
David Davis speaks to Nick Robinson about his decision
He vowed that the Conservatives, who are the official opposition and favourites to win at the next election, would continue the fight in the House of Lords.
Tory leader David Cameron paid tribute to his campaign on 42 days and insisted the party would not change its policy.
But he stressed Mr Davis' resignation had been "a personal decision, a decision he has made".
He said it was a "courageous" move and he hoped Conservatives would support Mr Davis's by-election campaign.
This resignation is quite extraordinary and without precedent that I can think of in British politics and means that politics is now utterly unpredictable
But he added: "We cannot put home affairs on pause and it is my job to ensure that we have a team that's ready for government."
Shadow attorney general Dominic Grieve, whom Mr Cameron has appointed as the new shadow home secretary, rubbished reports the Tory leadership was split on the issue of 42 days and pledged to repeal the measure if the party gained power.
Home Secretary Jacqui Smith said: "Faced with a crucial decision on the safety and protection of the British public, the Conservatives have collapsed into total disarray on what is their first big policy test since they have come under greater scrutiny.
"David Cameron must come clean on what has really happened and why David Davis has really resigned."
Lib Dem leader Nick Clegg, who also voted against 42 day detention, said his party would not be fielding a candidate in the by-election, after speaking to Mr Davis.
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