"Sunday play is being considered but a decision won't be made until tomorrow," a spokesman for the All England Club said Friday.
Play on the middle Sunday -- the traditional rest day in the Wimbledon fortnight -- last took place in 2004.
It has only happened on two other occasions: in 1991 and 1997.
This year's Wimbledon has been badly affected by the weather with long delays on Tuesday and Wednesday.
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Rain has seriously disrupted the schedules for the past four days. Organisers had to pay out refunds of between 50 percent and 100 percent on Wednesday's tickets for the open courts.
There were still 16 second round singles matches to complete on Friday -- the day usually reserved for the start of the third round.
Play on Centre Court, featuring Juan Martin del Potro against Stan Wawrinka was continuing under the roof, with multiple Wimbledon champions Serena Williams and Roger Federer due to play later Friday.
The last time matches were staged on the middle Sunday, in 2004, was prompted by a washout on the previous day.
In both 2004 and 1997, two days of play were completely rained off in the first week.
Before the roof was installed on Centre Court, play was extended to a third Monday in 1992, 1996, and 2000 in order to complete the finals, and memorably again in 2001 when Goran Ivanisevic, playing on a wild card, won the men's title on a Monday.
Since the Championships moved from Worple Road to the current Church Road grounds in 1922, there have only been six completely rain-free tournaments: in 1931, 1976, 1977, 1993, 1995 and 2010.