Nash said the deaths were "unacceptable" and encouraged the industry to come up with ways to minimise seabird bycatch.
One method, for example, was to use a hook-shielding device that only opens once it is 20m deep.
"If that can be commercialised, it would be fantastic," Nash said.
"The last thing any fishermen wants to see on the end of one of his hooks is a seabird. It is good to see the fishing industry has taken steps to do better, but more needs to be done."
But it is doubtful that commercial ventures would take up such methods unless there was a financial incentive to do so, and a spokeswoman for Nash clarified that the Government could not force the industry to use methods such as hook-shielding devices.
Conservation Minister Eugenie Sage also implored the industry to step up.
"Antipodean and Gibson's albatrosses are as endangered as kākāpō. We must do all that we can to protect them," Sage said.
In the 2016/17 year an estimated 579 seabirds were captured in surface longlines and 1846 in bottom longlines.
"The fishing industry needs to go over and above minimum requirements, and demonstrate leadership," Sage said.
She said the industry could look at weighing lines so they sink faster, setting them at night, using bird scaring lines, or simply stop fishing in certain areas or times of the year when there is a high risk of bycatch.
In November last year, the Government signed an arrangement with the Chilean Government to reduce bycatch in international waters between Chile and New Zealand.
The arrangement is expected to enable better exchange of information between New Zealand and Chile and support greater innovation to reduce albatross bycatch.
Since 2004, the albatross population has been declining at a rate of 6 per cent a year for males and 12 per cent a year for females, and the population may disappear within 20 years.
Sage said additional Government funding for conservation from Budget 2018 was being used to help reduce bycatch and improve fishing practices.
Fisheries NZ and the Department of Conservation were also working on the National Plan of Action to reduce the incidental capture of seabirds. Public consultation is expected to start soon.