According to a report in the Telegraph, chairmen have held informal talks about pausing the season for a short period in January after Manchester City's visit to Everton was postponed on Monday due to a raft of confirmed cases in Pep Guardialo's squad.
Sheffield United confirmed one player had tested positive on Tuesday but their trip to Burnley still went ahead at Turf Moor.
Blades manager Chris Wilder confirmed prior to his side's meeting with the Clarets that a number of backroom staff had also tested positive.
He said: "It is not ideal, but we have got to get on with what we have got and that's what we will do.
"We have just notified the relevant authorities that we have a situation at the football club, but I want to play football. We are not looking to get anything canned or anything."
Leeds United's record signing Rodrigo was forced to sit out the beginning of November after testing positive for the virus but the Whites have not been forced to rearrange any fixtures.
Earlier this season, Newcastle United's clash with Aston Villa was postponed due to a Covid-19 outbreak at the Magpies.
There were 18 positives from 1,479 tests on players and staff between December 21 and 27, the highest number recorded since top-flight clubs starting regular testing earlier this year.
There are also fears the third round of the FA Cup, scheduled for next weekend, could be heavily disrupted by the rise in case numbers.
Nine EFL matches have been postponed due to outbreaks with League One teams hit particularly hard, including both Hull City and Doncaster Rovers.
Rotherham United were forced to postpone their fixtures with Derby County and Middlesbrough while Bradford City have seen fixtures delayed due to coronavirus outbreaks amongst fellow League Two clubs.
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