Charles Lamb comes second in a primary in Edmond, Oklahoma, and will be on the final ballot in April.
Mr Lamb, who had been mayor since October 2011, put his name forward for re-election before he died.
When he passed away, it was too late to remove his name from the ballot or to add anyone else.
Were he to win on 2 April, a mayor would be appointed by Edmond City Council.
Mr O’Neil, who served one term as mayor from 2007-2009, said there were “people advocating for his election for their political reasons … they want to be mayor”.
A Facebook campaign supporting Mr Lamb was led by Edmond resident Michelle Schaefer, who referred journalists’ questions to councillor Nick Massey.
Ms Schaefer has reportedly said she would like Mr Massey to be mayor.
Echoing Mr O’Neil, Mr Massey also said it was an awkward situation, adding that had Mr Lamb not sought re-election, he would have run for mayor himself.
While he would be “honoured” to accept the position, Mr Massey said he would not be calling for people to vote for Mr Lamb.
However, he added: “If you like the direction the city has been taking over the last six, seven, eight years, you might consider voting for Charles and let the city council decide who to appoint.”
The late mayor’s son, Erik Lamb, said he, his sister and mother discussed the planned social media campaign before it became public.
“We were approached by outside people who asked if they would have our blessing and we gave it to them to go ahead and continue,” he said.