The issue will be on the agenda at next months shareholders’ meeting, where representatives from all 20 top-flight clubs will be present.
It is understood that the majority of the 20 clubs are keen to back the idea, with Watford believed to be one of the few to oppose the plan.
For the new rule to be introduced, at least 14 clubs must agree to the proposal.
The current deadline is August 31, almost three weeks after the season kicks off and is in line with many other European leagues.
Many believe the early closure would improve the league's integrity as none of the teams would be able to sign new players after the first match of the season.
Paul Clement, Swansea City manager, said "Our club would support that decision. The majority of clubs are in favour but maybe all have to be for it to go through. It could happen next year."
Others, however, believe Premier League clubs would be weaker as teams from other European leagues, including Spain and Italy, would still be able to sign players from England until their own deadlines.
Thierry Henry is among those who do not agree with the change and used the transfer saga surrounding Philippe Coutinho as an example to prove his point.
He told 'Sky Sports', "The problem is still the same. You can't buy but you can sell?"
"So Liverpool, as we all think, will lose Coutinho; he leaves you and you can't sign a replacement? They're trying to do the right thing but not in the right way. I don't personally agree with that one.