Monaco have been playing hard ball all summer in an attempt to prevent Lermar joining Benjamin Mendy, Tiemoue Bakayoko and Bernardo Silva in leaving the club.
However, after having two offers rejected by Monaco, the most recent being a bid of £45m, Arsenal are convinced the improved figure of £50m will be enough to get a deal over the line.
With the future of Alexis Sanchez still in doubt, Arsenal have insisted that Lemar is not a replacement for their star man. The Chilean is expected to return to pre-season training on Monday where he will discuss his future with manager Arsene Wenger.
It is understood that Arsenal have offered Sanchez a new contract worth £275,000-per-week, which would increase by up to 20 per cent, if the Gunners return to the Champions League.