He said issues such as longevity and acceptance of a nominee by other parties had to be taken into account.
He would expect the Maori Party to back the Government's choice for the role, although it was not a matter of confidence or supply. The Government would consult with all parties about the options - of which there were others in addition to Mr Williamson and Mr Carter.
Labour leader David Shearer said he was yet to be formally briefed, but he preferred Maurice Williamson of those whose names were known. He said Mr Williamson was quick on his feet, had an interest in the process of Parliament and had a sense of humour "which I think is a pre-requisite for the job."
He said Mr Carter had not shown the same level of interest in procedure, and he saw him more as the agriculture minister.
Mr Henare criticised the Maori Party for backing out of its agreement - a repeat of criticism he made last Friday when he also suggested someone should start another Maori Party which did not renege on its deals.
Maori Party co-leader Tariana Turia responded to that by saying they agreed to support Mr Henare very early on before being told the Government preferred someone else. She said she believed the Government should choose the Speaker.
Of Mr Henare's suggestion someone start another Maori party, she said it was a good idea: "He might like to revive the Mauri Pacific Party."
Although she and co-leader Pita Sharples had intended to meet Mr Henare to discuss it, his tweets meant that was now unlikely.