Mr Barilaro, currently deputy premier and Nationals leader in NSW, is considering the move over the weekend. He said the Coalition's failure in Gilmore showed the danger of three-cornered contests, and if party research showed he was the best candidate, both parties should get behind him.

"The truth here is this is going to be a tough seat. It's the bellwether seat and no party owns the seat," he said.

"If the Prime Minister wants to bolster his numbers in the parliament the best way forward would be a coalition unity ticket - one candidate taking on the Labor Party."

The Nationals won just 7 per cent of the vote in last year's election in the seat, with the Liberal candidate, truffle farmer Fiona Kotvojs, winning 37 per cent, and Mr Kelly 39 per cent for Labor.

But Mr Barilaro said he had won 52 per cent of the first preference vote last year in his state seat, and he insisted people vote for people more than parties.

Mr Barilaro is considered a national leadership contender, and current leader Michael McCormack gave a less than enthusiastic endorsement of him on Thursday, saying if he decided to put up his hand he had to get through preselection first and then the by-election, which would not be easy, with 100 years of history counting against governments winning by-elections.

"History has a habit of repeating itself," Mr McCormack said.

While Mr Barilaro was a good local candidate, others might also want to stand, he said.

"Anybody who wants to put up their hand, good luck to them."

Mr McCormack also touted his close working relationship with Bega Mayor Kristy McBain, who has been named as a Labor frontrunner.

Ms McBain did not respond to calls on Thursday. Nor did Senator Molan or Andrew Constance respond to queries about their candidacy. Mr Constance, also a state member of Parliament, built a national profile during the bushfires.

Asked about federal leadership ambitions, Mr Barilaro said he brought considerable experience to the federal party room as leader of the NSW Nationals and deputy premier "and if there's an opportunity for me to serve the Prime Minister in any capacity I'm always happy to do that".

But he expected to be a backbencher for some time - "and that's not a bad thing.

"You start your apprenticeship all over again."

If he did make the move, he would bring his "brawler, rogue, maverick approach", and would be "waving that National Party flag as hard as I can".

Former Liberal leader and Australian National University professorial fellow John Hewson said the by-election had multiple layers as a historically important seat and given Mike Kelly's big support base. Scott Morrison's popular handling of the pandemic and unpopular handling of the bushfires would play a part.

Dr Hewson dismissed Senator Molan, who he said had "scrounged his way" into the Senate after a preselection fight that upset many in his own party.

But he said Eden-Monaro jealousy guarded its local members, and both Mr Constance and Mr Barilaro had local popularity and profile.

"At this stage, you'd say there's an outside chance against the trend that the government might win," he said.

ABC election analyst Antony Green said everything was working against Labor except the bushfires. The biggest issue for Labor was the loss of Mr Kelly's personal vote in a seat where the margin was small.

"In a regional seat like this there's an awful lot of personal votes in there," he said.

"Absolutely everything about the contest says Labor will lose votes because Kelly's not the candidate, and the Nationals have a very high profile and popular candidate.

"So just those factors mean Labor's got some difficulty trying to hold the seat. The next question is do the bushfire and coronavirus have some impact, and that's impossible to measure."

It was an electorate in two parts and for the Nationals, the challenge was to garner votes among the electorate's coastal populations, where the party had no support base, he said.

If it came to a three-cornered contest with both the Liberals and the Nationals running candidates, the Nationals would have to devote significant resources to the coast.

But Australian National University politics professor John Warhust said Labor had a good shot at holding the seat with the right candidate.

"I would think it's a neck and neck contest and I'd maybe even have Labor ahead," he said.

He discounted Senator Molan's chances.

"I think Jim Molan would be foolish to stand. And at his age you'd wonder what he had to gain unless he had a high sense of his ability or really wanted to be in the House of Representatives. You would think the time he has left in the Senate would be enough."

Mr Constance would be a good candidate but given the "terrible battering personally" that he had taken during the bushfires and talk of him retiring altogether from politics, Professor Warhust said he was unsure whether Mr Constance would jump into a federal seat.

As for Mr Barilaro, he was a controversial figure, and it would be a "big bold step" for him to jump into a three-cornered contest and hope he could out-poll the Liberals. Such an outcome would be a real coup for the Nationals, which rarely took seats from the Liberals, he said.

The longer the by-election was delayed the more people were likely to support Labor, as the pain intensified from the economic crisis.

"If you're talking about a by-election in three months, much can happen in that time and possibly the government wouldn't be looking quite as confident," he said.

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"People might feel this is a chance to send a signal and with the government having a clear but still narrow majority, they can't argue that it's terrifically important.

"I think Labor would be a narrow favourite depending on their ability to come up with a good candidate."