The Premier League and Football Association tonight admitted there is no easy way to cut down the number of matches played.
Gordon Taylor, chief executive of the Professional Footballers Association, believes England's international ambitions are being scuppered by player burn-out and wants the sport's authorities to step in.
But Premier League spokesman Dan Johnson said: 'In 2004 the Premier League chairmen voted in principle for a winter break - but given the proliferation of fixtures it wasn't physically possible.
'We have to listen to the experts and it's a compelling argument - but we have to fit in a fixture structure.'
English football's top flight used to contain 22 teams, whereas now there are 20.
And Johnson explained: 'We've reduced them but other competition organisers have increased.
'There are more World Cup qualifiers and more Champions League games and unfortunately we are at the bottom of the pile in terms of when we can schedule our fixtures.'
As for reducing the number of teams in the Premier League further, he added: 'I don't think there is any appetite with the clubs to reduce the size of the league any further.'
Taylor cited the number of England players to be hit by persistent injuries this season as proof that their bodies are giving way under the demands.
And FA spokesman Adrian Bevington added on BBC Radio Five Live: 'We have a congested fixture programme - but when would we find time to have a winter break?
'When would it be and what would give? That's the difficulty.
'Something would have to give and there is no definitive answer as to what that would be.
'It can't be a decision the FA, in isolation, take. A number of stakeholders have a view and a role to play.
'It's very fair and sensible to have dialogue on this subject - the players clearly feel strongly about it and the thoughts of UEFA's medical staff provoke further debate.
'We want the best domestic game and best international side, so there are many factors which have to be taken into consideration.'