The Premier League will hold an emergency meeting on Thursday to establish plans for the remainder of the season.
Football League clubs will meet on Wednesday to discuss their own response to the coronavirus crisis.
It is believed Premier League clubs will have more information by then after a UEFA European football stakeholders meeting on Tuesday.
As it stands, no Premier League fixtures will take place until April 4 at the earliest, with the Football League also suspending all games until April 3.
The measure to postpone Premier League football was agreed at an "emergency club meeting" on Friday and in light of Arsenal boss Mikel Arteta and Chelsea midfielder Callum Hodson-Odoi confirming they had tested positive for the virus.
Some clubs believe the season can still be completed if games start again on April 4 but others think it is more realistic to start preparing for the possibility of there being no top-flight football until the start of next season in August.
One senior source at a Premier League club says he believes there is a "75 per cent chance this season will not be complete".
A step into the unknown
Analysis from Sky Sports News Chief Reporter Bryan Swanson
The next club meeting will sharpen minds on the longer-term impact of an unprecedented pandemic.
Premier League officials have been gathering expert medical advice and will put forward several options to all 20 clubs, with detailed financial and sporting implications.
By next Thursday, clubs will know more about the government's plans to ban mass gatherings, including football fixtures, and there should be a clearer understanding of the direction of travel following UEFA's emergency meeting on Tuesday about whether this summer's European Championships will be delayed.
Karren Brady's public "null and void" remarks may have bemused some clubs, and irritated others, but the Premier League will do all it can to complete the season. If Euro 2020 is postponed, it would free up space in June, but a lot will depend on further projected peaks for coronavirus in the UK.
The scheduled Euro 2020 final date of July 12 could be used as a marker for the end of the domestic season in Europe, too. That is the date when many top players will have hoped to compete for their country, at Wembley, before the start of a pre-season break.
Every Premier League club, and other European leagues, will have a preferred choice to deal with this extraordinary situation. An enhanced 22-team Premier League next season could put a further strain on an already congested fixture calendar. We can expect to see more ideas and options in the coming days.
The Premier League will want to avoid a public spat between its clubs. It prefers its decisions to be unanimous, as it was in deciding to suspend all games until April 4 at the earliest. If disagreements persist, two-thirds of clubs - 14 - are required to agree to any constitutional change.
Football authorities know, and understand, the game will always come second to public health. People's lives are at risk. This is a step into the unknown and, in an ever-changing world, nobody is quite sure what will happen next.
'Next coronavirus step divides Premier League clubs'
Analysis from Sky Sports News reporter Kaveh Solhekol:
Some clubs believe the season can still be completed if games start again on April 4 but others think it is more realistic to start preparing for the possibility of there being no top-flight football until the start of next season in August.
One senior source at a Premier League club says he believes there is a 75 per cent chance this season will not be completed.
Leading figures in the European game will decide what to do about this season's Champions League, Europa League and Euro 2020.
The Premier League would have extra time to complete this season's fixtures if Euro 2020 is postponed, but it remains to be seen if there is any appetite from players and clubs to play on into the summer - especially because the coronavirus infection is expected to build to a peak in the UK in May or June.
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Coronavirus - key sporting developments
- Premier League, EFL, FA Cup and Women's Super League suspended
- Champions League and Europa League matches postponed
- England Test series in Sri Lanka postponed
- Golf's Players Championship is cancelled
- F1 cancels Australian GP
- Coronavirus: NFL teams suspend travel
- NBA season suspended "until further notice"
- Euro 2020 likely to be postponed by UEFA
- ATP Tour suspended for six weeks
- PRO14 season postponed until further notic