Mr Nesbitt announced he is to quit as soon as a new leader is appointed, after the party's disappointing performance in March's election.
The UUP is now the fourth biggest party in the Assembly, having been overtaken by the SDLP for the first time.
Mr Swann made the announcement on social media on Friday.
He said he had "submitted papers" to seek election as leader of the UUP, after "receiving support from across the party, including members of our Westminster, assembly and local government teams".
The UUP is due to elect Mr Nesbitt's successor at its annual general meeting on 8 April.
There had been some speculation that UUP MLA Steve Aiken would throw his hat in the ring for the leadership, but he told BBC's Evening Extra he was supporting Mr Swann's bid.
"I think that it's excellent news and Robin will be an excellent leader. I signed his papers and I am fully supportive of him," he said.
"He is one of the few people who has the breadth and depth of experience. He is the chief whip, understands the system in the Assembly and has the vast majority of support from the party."
When asked why he had not put his own name forward, Mr Aiken said Mr Swann was the best candidate for the job.
Earlier in March Mr Swann told the BBC he was "not ruling out" a leadership bid.
He also said his wife Jennifer was a member of the Ulster Unionist Party and will have a vote at the AGM.
"If my name is on the ballot paper, I hope she will be voting for me."
The 45-year-old father of two was first elected to the Northern Ireland Assembly in 2011 and he represents North Antrim.
He chaired the Public Accounts Committee during the last assembly's mandate and is former president of the Young Farmers' Clubs of Ulster.
Mr Swann was one of 10 UUP members who won seats in the last Assembly elections, but he said the new leader did not necessarily have to be an MLA.
"Our party rules actually allow anybody who is in good standing, and a member of the Ulster Unionist Party, to put their name forward, so there is a misconception out there," he said.
Robin Swann has announced he has put his name forward for the Ulster Unionist Party (UUP) leadership contest following Mike Nesbitt's resignation.