Albury and other NSW councillors will remain in office for another year to counter the fallout from coronavirus.
The NSW Government Local Government minister Shelley Hancock has confirmed the postponement of elections due in September this year after days of speculation.
No decision has been made for Victorian councils which are also up for elections later this year.
But, Albury mayor Kevin Mack faces an election for his position in September when his present term expires and at least one other councillor, David Thurley, already confirming he is weighing up a challenge.
"Our present mayor has been one of the hardest working mayors of this city in recent years so he has a record and reputation that needs to be honoured," Cr Thurley said.
"I believe that I could be a good mayor and if I felt I had support I would stand and perhaps that would allow Cr Mack to have a breather and take a bit of downtime in what are very stressful times for our community.
"The situation we face now puts added pressure on our community leaders.
"I don't want to add to that and I am considering what is the best thing to do in the light of the recent announcements by the NSW Government and the unusual circumstances we find ourselves in because of the COVID-19 crisis."
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Cr Mack said: "It's fresh off the press and I am undecided (about seeking re-election as mayor).
"I will probably listen to my councillors and take it from there."
But in the event of Cr Thurley standing he would throw his support behind him for the top job.
"David has the experience and needs to be given a chance," Cr van de Ven said.
"He has had some great impacts with his role as chairman of the Murray Darling Association.
"I would love to be mayor again, but my personal circumstances at this stage with the COVID-19 impacts on business don't allow me to make that commitment."