In a frustrating quirk of the global schedule, the game's top two teams are not scheduled to meet until late next year, but the Press Association reports November 4 is now being examined as a possible window for an additional fixture.
"We are playing New Zealand in 2018", said an RFU spokesperson.
As it stands now the two leading sides in the global game are not scheduled to meet next until November of 2018.
If England topple Ireland in Dublin on St Patrick's Day weekend, they will have become the sixth team to secure back to back Grand Slams and will eclipse the All Blacks' record of 18 consecutive Test victories.
It is understood the Premiership clubs would have to give permission for the release of their players.
"We are playing NZ in 2018 - if the opportunity comes up for us to play them before then, we will pursue it".
A match against the current world champions would add an extra Test to the autumn series - scheduled at the moment to be just three games - and the Premiership clubs could demand a much bigger compensation payment.
Negotiations centre on the way revenues will be shared between the RFU and their New Zealand counterparts, as well as the RFU's need to compensate domestic clubs as the game would fall outside the designated global window.
England and New Zealand have not met since 2014, but the clamour for the dominant forces in the northern and southern hemispheres to clash is growing.
Owen Farrell is among the England players who could take that mantle from Barrett this year, yet both Jones and Hansen have conjured up a winning formula which is based around the team rather than individuals.
It was only when the RFU learnt that the All Blacks were in talks with the Barbarians about staging a fixture on November 4 that Ian Ritchie, the chief executive, approached the New Zealand Rugby Union.
However, asked if an earlier match could be accommodated, he added: "Not as our season is now programmed, no. It's highly unlikely at this stage", offered Tew when questioned about the possibility of the revised Test.
"We've got a very important series against the British and Irish Lions and that's a primary focus for us. I hope the experience gives rugby fans and novices alike an insight in to the way we train and the amount of hard work that is involved in being an global rugby player".