Mr Power's proposed changes soften the bill by enabling the court to allow jury trials for such charges in "exceptional circumstances", or ones considered to be complex or sensitive. The defendant would have to satisfy the court of such circumstances.
It would effectively mean that all cases could be heard by a jury, except where the offence does not carry a potential jail term.
Mr Power's amendment also waters down the ability of a court trial to proceed in the absence of the defendant - another clause that Labour, the Greens and the Maori parties find objectionable.
The bill as it stands states that the court "must" proceed if the accused is absent and does not have a reasonable excuse; the amendment would change the word "must" to "may".
Mr Power also suggested removing the provision around the right to silence and leaving it in the hands of the Rules Committee, a panel of legal experts.
The amendment would enable the defendant to remain silent without that being held against them at trial - even if the Rules Committee made it mandatory to notify issues in dispute before a trial.
But non-compliance could be an aggravating factor at sentencing, if found guilty.
However, there are only seven sitting days left this parliamentary term, and Mr Power is now seriously considering abandoning the controversial provision altogether.
Mr Power would not comment on details of the negotiations.
Labour's Justice spokesman Charles Chauvel said the bill was too important to be rushed through with a one-vote majority.
"If that is the attempt that's made, Labour would have to consider debating the bill extensively, and the Government would then have to decide if that's how it wants the last two weeks of Parliament to be spent."
In the dock
* Currently defendants can choose a jury trial for offences that carry a penalty of more than three months.
* The bill would push the threshold to three years, meaning most cases would be heard by a judge.
* Justice Minister Simon Power is proposing that the court decide to allow jury trials in "exceptional circumstances" for any case that potentially carries a jail term.