Councillors Andrew Whiley and Kevin Gilbert, both part of Mr Radich’s Team Dunedin ticket at the last election, told the Otago Daily Times they were backing Cr Sophie Barker for mayor.
Cr Brent Weatherall, the only other Team Dunedin member elected, said he did not intend to endorse anyone.
No other councillors, when asked by the ODT, voiced support for Mr Radich.
Cr Whiley said Mr Radich was not the best candidate to lead the city.
Asked for a comment on Mr Radich’s performance as mayor, Cr Whiley said "it proves that experience around the council table is extremely valuable, and understanding council processes".
"We need someone who can actually lead the city for the next three, six, nine years, because we've bounced around with what I perceive to be two one-term mayors."
Team Dunedin had done its job by moving the council "away from the agenda" of the previous council and mayor Aaron Hawkins.
"We could have achieved a lot more, but unfortunately some views sort of moved in a different direction — I'll leave it at that."
Cr Kevin Gilbert also said he believed Team Dunedin had achieved its goal in shifting trajectory away from the previous council’s "agenda".
Mr Radich had helped "move the dial", but a different skill set was now needed — "from what I can see, Sophie brings that", Cr Gilbert said.
"It has felt like others' opinions haven't necessarily mattered quite as much around the table and I think Sophie will be more encompassing."
Mr Radich did not comment specifically on the fact no councillor was publicly backing him, but said he agreed Team Dunedin achieved a significant shift away from the previous council.
"Unfortunately, it didn’t go as far as it might have," he said.
Although Team Dunedin was not fielding a group of candidates this year, he remained focused on practical and common-sense solutions, he said.
The council was diverse, with "divergent views", but had made decisions "generally" reflecting the community’s views, he said.
"Part of a listening leadership is that everyone’s voice gets heard, even those who have been critical from the very first day and continue to be so."
Cr Weatherall said Team Dunedin failed due to a "lack of unity and numbers elected", which was not the fault of Mr Radich.
"I would have been happy to endorse Mayor Radich if he was the mayoral candidate under a team I was involved in, however, that is not the case this election, so I do not intend to endorse him," Cr Weatherall said.
"From my now experienced personal perspective, teams don’t work."
He said Mr Radich had "worked hard and performed well" with the group of councillors that voters gave him.
"Diversity is great. However, it diminishes cohesiveness."
On Saturday, Cr Barker announced she would stand for mayor, joining Mr Radich, car dealer and Mosgiel-Taieri Community Board chairman Andrew Simms and fellow councillors Mandy Mayhem and Lee Vandervis in the race.
Cr Jim O’Malley said a mayor needed both the city’s interests at heart and the ability to give the rest of the council "the representative voice they have been elected for".
"Cr Barker has both those qualities — I’m not sure the rest of the field does."
Cr Barker said while it was still early in the election process, the support from councillors was "great".
"I’m very gratified by the respect and support of those councillors who show such faith in me."
Cr Steve Walker was critical of Mr Radich’s mayoralty this term, saying he had not been a united leader, had no vision for the city and was a poor communicator.
"It’s obvious from the voting records and the general discord around the table that both Andrew and Kevin have little confidence in the mayor’s directionless leadership."
"He may have run under the banner of Team Dunedin, but I think he left the word ‘team’ behind after he won the mayoralty."
Cr Walker said he was not running for mayor.
Cr David Benson-Pope said he was not running for mayor and did not endorse a candidate.
Cr Christine Garey also said she was not standing for mayor and would hold off endorsing any candidates until she knew who was standing.
Deputy mayor Cherry Lucas did not respond to questions on endorsing candidates but confirmed she would not be standing for mayor.
Crs Bill Acklin, Carmen Houlahan and Marie Laufiso did not respond to questions by deadline.
Candidate nominations open on July 4 and close on August 1. The election will be held on October 11.
