Nominations closed at midday on Tuesday, June 22.
Edwin Poots resigned last Thursday after a bitter internal party revolt, leaving the DUP looking for a third leader in a matter of weeks.
Sir Jeffrey’s name will go forward to the party’s electoral college, which is expected to meet on Saturday.
Candidates for the position had to submit their names to the party chairman, Lord Morrow, backed by a proposer and a seconder.
Following the meeting of the electoral college he will have to be ratified as party leader at a meeting of the DUP executive, which could take place next week.
He would become the party’s fifth leader in its 50-year history.
In a statement issued after nominations closed, Sir Jeffrey said he had the “vision to unite Northern Ireland and heal the divisions of the past”.
“The task ahead is great,” he added.
“I do not underestimate the challenge, but I know the overwhelming majority of people who live here want Northern Ireland to keep moving forward.”
However, his statement made no mention of the Irish Language and cultural act which the British Government said it will introduce at Westminster in October.
Lord Morrow said the past few weeks had been “difficult for the party and mistakes have been made”.
“Now is the time to move forward in a spirit of humility and mindful that our focus must be on serving the people whom we represent,” the peer added.
“I look forward to the challenges of rebuilding and reconnecting in the time ahead.”
The leadership vacancy arose when Mr Poots resigned after just 21 days in the position he took over from Arlene Foster.
He had defeated Sir Jeffrey in the party’s first leadership election on 14 May by 19 votes to 17.
Sir Jeffrey has already outlined his priorities and has said the number one issue is the dealing with the post-Brexit trade arrangements for Northern Ireland, known as the Protocol.
“If elected, I will ensure that the government doesn’t just listen but recognises the need to take decisive action to deal quickly with the protocol,” he said on Monday.
He has also insisted he is the right person to lead the party to victory at the next assembly election.
Mr Poots’ resignation came after he agreed a deal with Sinn Féin and the Westminster government to ensure Paul Givan became the North of Ireland’s new First Minister.A majority of DUP members in the Stormont assembly wanted to delay that process and Mr Poots faced an internal revolt at a party meeting.
A vote which took place at the meeting saw the vast majority of DUP representatives present oppose Mr Poots nominating Mr Givan as first minister.
Mr Poots was then summoned to a meeting of DUP officers on Thursday evening, after which he announced his resignation.
Mr Donaldson is expected to resign his Lagan Valley seat in the autumn to become First Minister and lead the party from Stormont.
Tags: