FA general secretary Alex Horne admitted the position, which has been shared between the four British associations since 1946, was of limited value - it has been the target of resentment in other parts of the football world for years.
Horne, in Istanbul for the UEFA Congress, said the FA's priority was to maintain their presence on the law-making body known as the International FA Board (IFAB).
He said: "We are open-minded about it. If you look at it objectively the vice-presidency is quite anachronistic and I wouldn't fight tooth and nail to keep it if I'm honest.
"I'm not sure there's a great benefit in keeping it - if we give it up in the right way there will be a little bit of goodwill, and it could mean an extra UEFA seat (on FIFA's executive committee).
"As far as IFAB goes that's much more important to us - the status of IFAB and the conservatism around controlling the laws of the game and the sanctity with which we believe the laws should be protected.
"Contractually we do believe we have a position on this - we entered into IFAB first and FIFA then joined us (in 1913).
"We feel a lot more strongly about IFAB and if there are sensible conversations to be had about composition then we will listen to them."
The post has been targeted as part of the FIFA reforms, which will also look into the process by which all FIFA vice-presidents are elected by their confederations.
The Scottish FA also accept that the vice-presidency may have to go.
SFA chief executive Stewart Regan said it would be wrong to cling to the past at the same time as pushing for change.
Regan said: "Within the Scottish FA we have undergone our own governance review in the last 18 months and have gone through very similar processes about what needs to change to modernise the game.
"We have done it in our own part of the game so what we can't do is try to cling on to bits of the past at a global level.
"We welcome the reform that FIFA are putting forward - we were one of the countries that suggested the need for a governance review.
"On the one hand we have requested a formal governance review so we can't sit back and say we want one part of the cake but not the other, so we will discuss the outcome once it happens."
The current FIFA vice-president is Jim Boyce from Northern Ireland and if the position is abolished it is likely to be when his four-year term ends in 2015.
Before him, England's Geoff Thompson held the position for four years, succeeding Scotland's David Will who was in place for 17 years.
Will took over in 1990 from Northern Ireland's Harry Cavan, a legendary FIFA figure who had been the British vice-president for 30 years.
Wales have never held the position and their FA too believe the time may have come for it to be scrapped.