"I'm not going to make any predictions.
"Rex was a fantastic chair - he's left an impressive legacy."
Barker, if he runs, faces potential competition from former Tukituki MP and first-time councillor Craig Foss, who is considering putting his hand up for the leadership position.
"Rex standing down was a huge surprise, and it's taken a little while to digest.
"The work has to continue, though, and electing a new chair is a key part of that."
He said he was "very grateful" to be working with his fellow councillors and said he could see any of them being suitable for the chair's role.
Foss said he hoped to see good candidates put their hand up for Graham's vacant seat as "it's such a cool job".
Fellow first-time councillor Hinewai Ormsby was also considering making a bid for one of the leadership positions, adding she was passionate and saw herself as a long-term candidate who had strong roots in the community.
"We are still figuring that out.
"We are really lucky that we've got a good bunch of councillors."
She is chairwoman of the environment and integrated catchments committee, which plays a key role in the management of council's core environmental activities.
One regional councillor who won't be throwing his name in the ring is Neil Kirton, who told Hawke's Bay Today he was more focused on his committee roles and his constituency - but joked he would happily take the chair's car park if it became available.
The more hotly contested race may come in the by-election, with nominations now open until midday on July 15.
If more than one nomination is received, an election will be held by postal vote using the First Past the Post (FPP) electoral system. A postal vote would take place from August 19 until midday on September 10.
Tom Belford, Peter Guerin, Tim Gilbertson and Mary O'Neill were the four highest-polling unsuccessful candidates for the Hastings seat at the 2019 elections, each getting more than 4000 votes.
Belford and Gilbertson have past council experience but it is unknown if they will stand again. O'Neill has over the past year made headlines over her string of court battles with her Napier neighbour.
Guerin, who narrowly missed out during the last election, said he was thinking about trying again but was still undecided.
"The issues I was campaigning on haven't gone away."
Former Tukituki MP Lawrence Yule has the political experience to build a case for inclusion on the council. He said he had not yet made any decision.
Green Party candidate for Tukituki Chris Perley was also undecided.
Fellow Green Party candidate for Napier James Crow ruled out running for the seat, saying his focus was central government and his own patch in Ahuriri.
"It sounds like some good people will [run], though."
Nomination forms are available from the regional council's Napier office or online - they must be received by midday on July 15.