The East Antrim MP had been considering putting his name forward after Nigel Dodds ruled himself out.
Finance Minister Arlene Foster is the only person so far to declare an interest and has the backing of a majority of the party's most senior elected representatives.
Nominations to become the next leader of the DUP close later on Wednesday.
Peter Robinson announced in November that he was standing down as party leader.
On Tuesday, Mrs Foster said she was "very humbled" by the support she has received from party colleagues.
Mr Wilson thought long and hard about his decision and spoke to among others, party colleagues Mrs Foster and Mr Dodds.
In the end he felt it would be in the best interests of the party that he did not put his name forward.
He felt he wanted to make sure there was a smooth transition and that means almost certainly now that Arlene Foster will be the new DUP leader and the new first minister at Stormont.
She said she looked forward to leading the DUP, if that was the party's wish.
Mrs Foster said she had hoped to work with Mr Dodds as a team.
"We will still hopefully work together as a team and that is certainly my wish for the future," she said.
In a tweet on Monday night, Mr Robinson said he had "received a valid nomination" from Mrs Foster for the post of DUP leader.
"Arlene's nomination was submitted with the support of over 75% of those entitled to vote in the electoral college," he added.
Sammy Wilson has said he will not be running for the DUP leadership.