Having been the Potters' third choice 'keeper last season, the veteran Englishman is said to be weighing up his options and deciding whether to seek first-team football elsewhere
The 38-year-old was third choice last season behind Asmir Begovic and Thomas Sorensen, and stated on his Twitter account that he is assessing his options for next term.
However, Stoke’s director of football John Rudge told the Stoke Sentinel: "We have offered Carlo a contract.
"He would like to be playing, but he is a good type who has done well for us while he has been here.
"We hope he stays because he is an experienced player who gives us a good battle for places."
Nash has had a nomadic career in English football, with spells at Manchester City, Preston North End and Everton amongst others before arriving at the Britannia Stadium in 2010.
However, he has only made one appearance for Stoke since joining them, playing in their 2-1 win over Shrewsbury Town in the Carling Cup in August 2010, and he may look to find first-team football elsewhere.
Should Stoke fail to agree a new deal with Nash, they may look to youngster Dale Eve to provide cover in goal.
The 17-year-old agreed a two-year professional contract at the end of the season, and became the club's youngest-ever full international last November when he played for Bermuda in a World Cup qualifier against Barbados at the age of 16.