I've no idea if there will be a normal transfer window or not, but the date on which it opens is midnight on the last day of the season, not 10 June.
According to page 219 of the Premier League Handbook 2019/20, in the section "Transfer Windows"
V.2. The Summer Transfer Window in any year shall:
V.2.1. conclude at 17:00 on the Thursday before the commencement of the
relevant Season or at such other date and at such other time as the Board
may determine in its discretion (which will only be exercised in the event of
agreement by a simple majority of Clubs on an alternative date and time);
and
V.2.2. commence either: (1) at midnight on the last day of the Season; or (2) at
midnight on the date 12 weeks prior to the date on which it is to conclude
(in accordance with Rule V.2.1, above), whichever is the later.
As the 2020-21 season is still due to begin on Saturday 8 August 2020, then (2) above isn't applicable.
Like I say, all bets are off as to what will eventually happen, but thems the rules