According to the Daily Telegraph, the Potters are expected to make a subsequent bid, with a view to making a deal before the end of July. Villa are believed to want at least £10m for the defender, but their parlous financial situation means that they need to raise funds urgently.
Stoke are the first club to make an actual bid for a player, but it is highly likely that others will follow suit, as they will hope to take advantage of the stricken Midlands club.
Manager Steve Bruce has openly admitted that Villa need to offload some of their players in order to comply with Financial Fair Play regulations. He said: "We don't want to lose Grealish, [Jonathan] Kodjia and Chester but the inevitability is we might have to so let's get ready for it."
Bruce's narrow failure to secure promotion from the Championship has proven exceptionally costly, as it has left Villa with a £40m black hole.
Despite Villa's financial needs, their fans will no doubt be furious about the potential loss of Chester, who was an integral part of the Wales side who stunned Belgium to reach the European Championship semi finals in 2016.
He was also the fans' player of the season in the last campaign, which ended with a 1-0 defeat to Fulham in the play-off final.
Chester is now 29 years old so is likely to have several years of top level football ahead of him. He is also vastly experienced, with 269 appearances in English league football and 29 international caps.