The party is also looking to win the support of one of the nationalist parties in a bid to keep alive constituency boundary reforms that would improve Tory chances of securing an overall majority at the 2015 election. Prime minister David Cameron fears Labour and Liberal Democrats will combine to defer the boundary reforms until after the election.
Tory MPs aware of the discussions in the party point to a deal on cheap air passenger duty for long-haul flights from Belfast, announced last week, as the kind of offer that may persuade DUP MPs to back the boundary reforms. The DUP said, however, that the duty deal was a longstanding agreement, had no relationship with any other issue and denied any wider deal was under discussion.
Plaid Cymru MP Jonathan Edwards has already said his party would be open to discussing a deal with the UK government over boundary reform. The draft plans see the number of Welsh MPs cut from 40 to 30.
But Edwards said any deal would have to involve a significant transfer of powers to the Welsh assembly and a new Government of Wales Act in order to offset the reduction in numbers of MPs.
He insisted he was speaking in an individual capacity, saying that no direct contact had been made between his party and No 10.
The crisis over the boundary reforms has led to some of the most bitter exchanges between the offices of Cameron and Lib Dem leader Nick Clegg since the coalition was formed, with both sides accusing the other of bad faith. Cameron has been pressed at meeting of the Tories' 1922 backbench committee to force the boundary changes through before the election.
The crisis started in the summer when Clegg reacted to the defeat of his House of Lords reform plans by announcing his party would no longer support boundary changes designed to make the size of constituencies more equal. Both houses were due to vote on the Boundary Commission proposals next year once the redrawn boundaries had been finalised.
Cameron responded to Clegg's move by saying he would defer the issue until next year, giving himself time to look for a way of pushing through the changes, seen by some as vital to Cameron being able secure an overall majority at the next election.
But Cameron's timetable has now been pre-empted by the surprise decision of the Lib Dems to use the electoral registration and administration bill, introducing individual voter registration, to change the date for the report from the Boundary Commission on parliamentary constituencies from "before 1 October 2013" to "not before 1 October 2018".
The amendment has the support Lord Hart of Chilton, a senior lawyer and former adviser to two lord chancellors, Lord Kerr of Kinlochard, a former Foreign Office permanent secretary, Lord Rennard, a former Lib Demchief executive, and Lord Wigley, a former leader of Plaid Cymru.
If put to a vote in the Lords, the amendment would be carried. Conservative whips would then have to try to overturn the amendment in the Commons without Lib Dem support, the first time the coalition whip would have split in the Commons on a government bill.
The coalition has an overall majority of 86 in the Commons, but the Conservatives are three short of an overall majority if Lib Dem MPs vote with Labour. The Conservatives are trawling for support among the eight DUP MPs, three Welsh nationalists and six Scottish nationalists.
Nigel Dodds, deputy leader of the DUP, said he had not been approached by the Tories on the issue of voting on boundary reforms and denied that the partial devolution of powers on air passenger duty had been part of any deal. Air passenger duty on direct long-haul flights departing from Northern Ireland airports falls to to zero with effect from January 2013. But Dodds said: "That was agreed by George Osborne last year."
The Scottish National party said it would not hold talks on a deal, but Conservative sources said they might vote for the government if the mood music was right over the referendum on Scottish independence.
The issue has become so explosive that the Conservative leader in the Lords, Lord Strathclyde, has twice deferred votes on the Hart-Rennard amendment in the Lords. He has also claimed that the amendment is out of order since it is not relevant to the bill but, in the Lords, it is up to peers, not the clerks, to determine what is permissible for inclusion in a bill and there is no obligation on the peers to take the advice of clerks.
In public, the three parties say they will oppose a deal.
The Conservatives cannot leave the issue on hold in the Lords indefinitely because they want the individual voter registration bill to become law in time for the changes to be implemented for the 2015 election.
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