The club is concerned that visibility for spectators is poor in comparison with its neighbouring No 1 court, and in comparison with top stadiums in Australia, France and the USA.
The club will discuss the move after the first grand slam event, the Australian Open.
Martin Ayliffe, the All England club's head of projects and development, said: "There is nothing concrete, but it is one of many issues that will be at the forefront of discussion after the Australian grand slam. There are lots of things the club needs to look at next to the other grand slams."
The upgrade would centre on a redesign of the bowl around the court that seats the spectators.
A source close to the club said the profile of the seating slope meant that people sitting at the back of the stands often had their view blocked.
The front seats are better than they need to be, the back seats much worse
The source said: "There are infinitely better sight lines in No 1 court – the club will have to tear down the concrete steps and put it back differently.
"The front seats are better than they need to be, the back seats much worse."
Sources suggest that it was originally thought that the court's capacity would have to be reduced to 11,000 to accommodate the improved sight lines, but subsequent thinking suggests that it might be able to retain the present 13,000 seats. The redevelopment would largely consist of precast concrete work.
Any development work would not start until July at the earliest.
All build programmes at the club have to take no more than nine months, as the club could not afford to miss Wimbledon fortnight in June.