Arsene Wenger has warned that closing the transfer window before the start of the Premier League season would still be problematic.
Sources indicate that the "majority" of top-flight clubs now favour an earlier closure, and the issue could be voted on during their next shareholders' meeting in September.
Changes to Premier League rules of this magnitude require a 14-vote majority.
It is understood talks will take place between every club next month about whether to change the deadline of the current window, which closes almost three weeks after the first game of the season.
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Southampton's Virgil van Dijk is one such case with the Dutch worldwide defender training on his own as several higher profile clubs chase his signature, while Icelandic star Gylfi Sigurdsson is another who is seeking a £50 million (55m euros, $65m) move to Everton.
Many Premier League managers, however, have grown tired of the window and the impact it can have, nearly three weeks into a domestic campaign.
But Liverpool manager Jurgen Klopp would still have to fight to keep Philippe Coutinho out of Barcelona's clutches until 11 p.m. on August 31 unless clubs in La Liga were to follow suit and impose its own preseason transfer deadline.
And Swansea's Paul Clement was the latest to do so over the weekend.
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It follows a succession of bids from Leicester City for the player, the last of which was said to be worth £15 million. Evans came through the ranks at Manchester United , but left for West Brom two years ago.
The 2017/18 Premier League season has finally started and boy have we been treated to some wonderful action already.
If the window is shut before the season starts, everything is sorted out and we can get on with the football.
"It would have helped us this year (if the window had shut when the season began)", the German said. It makes sense that when the season is starting, planning for the team is over.
Premier League clubs could lose players, and then be unable to sign anyone to replace them, or any players to reinforce their sides once the season has begun.
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Swanson added that a "majority" of clubs are in favour of the proposal, and the talks―and a possible vote―could take place next month.