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Esson had been Aberdeen's regular number one since he replaced David Preece in the side in December 2004 but recently lost his place in the team to Jamie Langfield.
Langfield has since proved himself with a number of assured performances, but Esson has still been offered new terms at Pittodrie.
Dons boss Jimmy Calderwood told the Press & Journal: "I think Ryan will be getting back to me in the next day or two. I would certainly hope that he comes in with a positive answer."
Calderwood is also understood to be thinking about loan moves for Scott Muirhead, Jamie Winter and John Stewart, but with a limited squad available and several injuries to key players this season Calderwood is wary of letting people leave Pittodrie without replacing them first.
Stewart is wanted by Queen of the South, following discussions between Calderwood and former Dundee United binman Ian McCall a month ago.
Calderwood said: "I will not be putting any players out until I get other players to the club first. There are different avenues we could go down just now, but it depends on who we get in."