The Blues have been pushed out to 7/2 second-favourites, sitting ahead of Liverpool (5/1), Everton (6/1) and Man United (8/1) in the betting.
Emery made Ramsey one of his five captains before the start of the season, when he made it clear he wanted the 27-year-old to extend and commit the prime of his career to the club.
For Ramsey, after so long with one club, now may be his best shot at getting one last big contract within Europe's upper echelon of leagues.More news: You Are My Friend First Look Reveals Tom Hanks as Mister Rogers
'Each player, they have this individual situation with his contract and this is one thing - for me today is not the day to speak a lot.
Initially, it seemed the former Paris Saint Germain boss was struggling to get his message across as Arsenal were swept away by champions Manchester City on the opening weekend before squandering a two-goal comeback in a 3-2 defeat at Chelsea.
"For me, every player is very important with their relation with the team", he explained.
Emery was also asked about another midfielder, Mesut Ozil, and the question of the player's decision not to meet with Joachim Low when the Germany coach visited Arsenal's training ground this week.More news: Read Christine Blasey Ford's written testimony to Senate
"Maybe I will someday, who knows what will happen in the future?"
"I think no [it won't change his performance]", he said.
Sevilla's Ever Banega reportedly emerges as a top target for Arsenal as they prepare for the impending loss of long-serving midfielder Aaron Ramsey. "He's been at Arsenal for ten years and they've let his contract run down".
Arsenal are keen to earn a fee from the player's departure and are prepared to listen to offers for him during the January transfer window. They are professionals for 10, 12 or 15 years, but not more, and they need to take the best performance and also work very hard in these years for his future and his family's future.More news: Google quietly started logging people into Chrome without their consent