City manager Pep Guardiola reportedly refused a swap deal involving Alexis Sanchez and Sterling during the final days of the summer transfer window, but the Gunners remain interested in taking the England international to the Emirates.
Sanchez is almost certain to leave Arsenal, but there is still some uncertainty as to when that will happen. City are believed to be confident of securing the player on a free transfer next summer, but there is still a possibility of an offer being placed in the January transfer window for the Chilean star.
The Daily Mail reported earlier this month that a £20m bid is being considered by City, but it seems Arsenal would like Sterling to still be included as part of any deal.
They are said to be keen to replace Sanchez if the 28 year old fails to perform between now and January, but it remains to be seen whether the sale of Sterling in exchange would be acceptable to Guardiola.
The winger has scored for City in his last three Premier League outings, although his game time would be limited if Sanchez were to arrive at the Etihad stadium. Much will depend on whether Gunners boss Arsene Wenger is willing to keep Sanchez until the end of the season and allow him to leave for nothing.