The Gunners are in the market to add a goalkeeper to the ranks this month as they prepare for the possibility of allowing Bernd Leno to depart.
Turner is wanted at the Emirates Stadium and Arsenal have slapped a bid in for the shot-stopper, who is on the books at New England Revolution.
The Premier League giants are hoping that an offer of around £5m will be enough to do the trick.
Turner is the United States’ number 1 goalkeeper and it remains to be seen if the Gunners’ proposal is acceptable.
The goalkeeper is 27 years old and New England Revolution are the only club in the MLS he has turned out for in his career so far.
Arsenal have Aaron Ramsdale as their number 1, with the custodian quickly establishing himself at the Emirates following a summer move from Sheffield United.
Turner will be looking to challenge Ramsdale for the number 1 spot at Arsenal if he does make the move to north London.
