The Premier League wants clubs to resume full contact training as early as next week, according to reports.
The Times claims top flight bosses are now hoping to step up the phased return to full training with a view to restarting in mid-June, but could face push-back from players.
All 20 clubs resumed group training this week following a unanimous vote, though strict guidelines remain in place which limit the number of players involved in joint drills and the time each member can spend on the training field.
A June 12 start has been described by the top flight's chief executive Richard Masters as a "staging post", but for the Premier League to meet that target clubs must step up training quickly.
Once the government has issued guidelines over full-contact training, the Premier League will produce a number of presentations for players and club staff with several still sceptical over plans to make such a swift return.
It is understood some Watford players opted against training this week after three positive tests, while it is unclear when N'Golo Kante will next train with Chelsea after the Frenchman was given compassionate leave over coronavirus fears.
There is particular concern among players from BAME (Black, Asian and minority ethnic) backgrounds, with evidence Covid-19 is disproportionately affecting people from those communities in the UK.
Troy Townsend, father of Crystal Palace's Andros, told The Times: "I've always thought the highest level of the game should have better representation and probably that is now being highlighted because maybe those concerns would go up the agenda. Maybe certain circumstances and decisions would be made in a different light."
