This claim is supported by FourFourTwo magazine, which purports to list "The 40 oldest clubs in the Football League". (Macclesfield Town, formed in 1876, is tenth on the list.)
Stoke City FC's claim to have been founded in 1863 is promulgated by the inclusion of that year on the club's badge. But there is no documentary evidence to support this claim. Stoke's first recorded match was played in 1868. Essentially, it does appear that a club was formed in 1863 by a group of Old Carthusians who were working as apprentices at the North Staffordshire Railway works in Stoke; but there is no evidence to link this club to Stoke Ramblers, whose formation in 1868 is the earliest event to which Stoke City can trace its history.
I had my doubts about this answer anyway, as I believed Stoke to have gone out of business some time in the Edwardian era. This is in fact confirmed by Wikipedia: "Continuous history from reformation in 1908 following the wind up of the original club."
The Nottinghamshire Live website reported on 9 May 2019 that there were doubts about Stoke's claim and that the EFL were investigating. If Stoke's claim was rejected, the title would simply make the short trip across the River Trent from Meadow Lane to the City Ground, home of Nottingham Forest FC. Forest switched to playing football instead of shinney – a form of hockey – in 1865.
Forest's own website reported on 10 May that the EFL had settled the dispute in their favour. (The club tried not to sound too pleased at gaining the title at the expense of its nearest rivals.)
Wikipedia (on the page previously cited) agrees: "The EFL confirmed that Nottingham Forest are now the current oldest member club due to the lack of evidence to support Stoke City's claim of having formed in 1863."
© Macclesfield Quiz League 2019