Wales, Ireland, Scotland and Italy want the present format of five rounds played over seven weeks, a system in place since 2000, to remain unchanged.
England are reported to favour all nations agreeing to a six-week schedule with one rest weekend in the middle. If there is no consensus, then a change setting England on France on a different schedule is also being discussed.
The proposals have gone through to the RFU's Professional Game Board, The Times reported.
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The RFU is looking at the changes after World Rugby approved a new global calendar from 2020, despite opposition from Scottish and Welsh governing bodies on player welfare concerns.
As part of the changes, England's summer tours are set to move to July, while the English Premiership are planning to begin the 2019-20 season in September, as usual, but end in June.
(Reporting by Shrivathsa Sridhar in Bengaluru; Editing by Christian Radnedge)
