World football's governing body have come under increased scrutiny in the last week after controversy around a FIFA Ethics Committee report into the bidding process for the 2018 and 2022 competitions.
Published by Hans-Joachim Eckert, the chairman of the adjudicatory chamber of the committee, the report cleared Qatar and Russia of any wrongdoings, claiming "various incidents which might have occurred are not suited to compromise the integrity of the FIFA World Cup 2018/2022 bidding process as a whole".
That saw former United States attorney Michael Garcia, who led the investigation, to claim the report was an "erroneous representations of his facts" and appeal the findings.
Bernstein believes FIFA must now undergo major reform to restore confidence, and UEFA's members should consider taking a drastic stance if that does not happen.
"England on its own cannot influence this - one country can't do it. If we tried to do something like that we'd be laughed at," he told BBC Sport.
"I think England within UEFA undoubtedly have the power to influence FIFA, but to do so they would have to consider withdrawing from the World Cup, the next World Cup, unless proper reform - including Mr Blatter not standing [for a fifth term as president] - is carried out at FIFA.
"If I was at the FA now, I would do everything I could to encourage other nations within UEFA - and there are some who would definitely be on side, others may be not - to take this line.
"At some stage you have to walk the talk, stop talking and do something.
"It sounds drastic but frankly this has gone on for years now, it's not improving, it's going from bad to worse to worse.
"There are 54 countries within UEFA. There's Germany, Spain, Italy, France and Holland - all powerful. You can't hold a serious World Cup without them. They have the power to influence if they have the will.
"I think it would be supported by responsible people in this country and actually by the public, because although the public of course want to see England play in the World Cup, I'm sure they would be prepared to take a degree of sacrifice to see that happen."
