The Lagan Valley MP is not expected to be opposed following a bruising party leadership contest last month.
Nominations opened on Saturday and are to close at noon on Tuesday.
The next leader will be ratified by the DUP’s central executive committee on Saturday.
It comes following a turbulent two months for the DUP that saw former leader Arlene Foster resign after an internal heave against her and her successor Edwin Poots follow suit after he was also fatally weakened by a party revolt.
His resignation on Thursday night came after just three weeks in the post.
It was prompted by his decision to press ahead with reconstituting the Stormont Executive alongside Sinn Fein, despite a significant majority of his MPs and MLAs being vociferously opposed to the move.
First Minister Paul Givan remains in post. It has been reported that he has been urged to resign when a new leader is in place.
Once Mr Givan resigns, deputy First Minister Michelle O’Neill will also be removed from her post, triggering a seven-day period during which new office holders must be nominated or a fresh Assembly election will be called.
Sinn Fein Finance Minister Conor Murphy said his party has no demands before renominating a deputy First Minister.
There is speculation the DUP may demand action from the UK Government over the Northern Ireland Protocol before nominating a new First Minister.
Mr Murphy described the DUP as “being at a crossroads”.
“They have to decide whether they want to be in a working Executive with the rest of us on the basis that they agreed to as a party – they signed up to this – or they want to frustrate things and prevent things, and create a very difficult working arrangement and ultimately threaten the future of the institutions,” he told BBC Radio Ulster.
“We have a lot to be getting on with, we have a pandemic that is still active, we need to see that out, we have huge economic challenges ahead of us, we have public services that have been starved by years of austerity cuts and we have a responsibility on behalf of the whole of society to deliver on these issues.”
Mr Murphy denied his party had made additional demands over Irish language legislation last week to nominate Ms O’Neill as deputy First Minister.
He said the party was “staying faithful” to the New Decade New Approach deal.
“We have said we are not pushing for anything else, we want to just see the things that were agreed under NDNA implemented in the way they were agreed to,” he added.
Sir Jeffrey narrowly lost to Mr Poots in the first leadership contest in the DUP’s history.
Chastened by that bruising campaign, the party hierarchy’s preference is for an uncontested appointment this time round, though it remains to be seen whether Sir Jeffrey will face any challengers.
During the last campaign Sir Jeffrey vowed to quit as an MP in Westminster to return to the Assembly and take up the role of First Minister.
If that were to happen, it would mean a Westminster by-election in Lagan Valley, a seat held by Sir Jeffrey since 1997.
One possibility is that he could seek to replace Mrs Foster as the MLA for Fermanagh and South Tyrone when she steps down.
However, with the DUP currently in such turmoil there are doubts whether the party would want to voluntarily trigger a by-election in Lagan Valley any time soon.
Sir Jeffrey’s commanding 19,000-vote majority in Lagan Valley in the 2017 general election was trimmed to 6,000 votes at the 2019 poll following a surge in support for the Alliance Party.
Another option could see Sir Jeffrey wait until just before the next Assembly election, scheduled in May 2022, before he resigns his parliamentary seat.
In that scenario he might appoint a temporary First Minister to fill the role in the interim.
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