The Football Association has rejected the Premier League's initial plans to play matches abroad.
Lord Triesman says the current proposal is unsustainable
The "39th game" proposal was discussed at a board meeting on Thursday where the FA told the Premier League to rethink its current plans.
"We haven't got what I would call a sustainable plan in front of us," said FA chairman Lord Triesman.
"The whole of the process requires some fresh thinking and some substantive answers to the questions I've posed."
He added: "If the Premier League have other proposals, I hope they come back with them relatively quickly."
However, Triesman has agreed to travel to Zurich with Premier League officials on 28 February to discuss the plan with Sepp Blatter, president of football's world governing body Fifa.
We have to have to try to ensure the football family as a whole feels more content with any proposal going forward
Lord Triesman, FA chairman
But BBC sports editor Mihir Bose understands that Premier League chief executive Richard Scudamore, who unveiled the radical plans for a round of "international" matches to be held in five cities across the world from 2011, will not receive the warmest of welcomes from Blatter.
"I am told that Richard Scudamore will get a very, very frosty response," said Bose. "Fifa is not at all happy with the Premier League, their blood is up on this."
Triesman said the Premier League had failed to answer the four issues he had outlined in Parliament earlier this week and that the rest of the board agreed with him.
"Firstly there are serious problems of congestion in the season," he said.
"I make it very clear that we would want to make sure that all of our own competitions can be played successfully, and that there isn't the usual overstretch that you get at that time of year which can have an impact on the England side.
"Secondly, I think that we have to make sure that our international relationships are in good shape; there are very many reasons for that, and the 2018 World Cup is one of those reasons.
Blatter is expected to give Scudamore's plans a frosty reception
"Third, there is bound to be continuing concern about whether the 39th game would change the symmetry of the competition or would introduce unfairness as it is perceived in the competition.
Of the potential threat to the 2018 World Cup bid, Bose added: "I understand Lord Triesman told the board meeting that England plan to launch their World Cup bid in May at the Fifa congress in Sydney.
"This would tactically be a shrewd move and the last thing he wants is to go to Sydney with plans to launch the bid and the questions he is asked are not how good are England and how they can stage the World Cup but about this 39th match."
Since Scudamore announced his plans there has been sporadic condemnation from football associations and cities touted as potential venues.
Triesman has noted the initial hostility and is keen to ensure any proposals meet the approval of all those concerned.
"It's also important to see that we're in a very difficult climate; people are very hostile and we've seen a great deal of that," he said.
"We have to try to ensure the football family as a whole feels more content with any proposal going forward."
On Tuesday, Scudamore said the plan was still alive, despite the widespread opposition.
He was also keen to meet with Blatter, who initially insisted the scheme would never get off the ground while he was in office.
"Clearly, we are not going to take this forward if it doesn't meet with some form of acquiescence from Fifa," said Scudamore.
"We were told we would have the opportunity to talk to the Fifa president about our proposal.
"I do not see how there can be speculation about the possible impact on the bid to stage the 2018 World Cup until this meeting has taken place."
That crucial meeting is set to take place in Zurich on Thursday 28 February.