The 21-year-old France international has been a target for the Gunners for much of the summer, with Liverpool also interested in him.
It is understood Arsenal made a late push to sign Lemar after Manchester City made an improved offer for their forward Alexis Sanchez.
While Gunners manager Arsene Wenger has been insisting for months that Sanchez, whose contract expires at the end of the season, would not be sold, it is believed the Chile forward would have had a chance of a move if Arsenal could have signed a replacement.
Some reports suggest Arsenal offered as much as Â£92million for Lemar, who joined Monaco from Caen in 2015.
But, with the deadline set for 11pm, it is understood the club are now not expecting to get the deal over the line on time.
That ultimately means Sanchez is likely to stay put, even though it is understood City launched an improved bid earlier on Thursday.
Arsenal had already lost Alex Oxlade-Chamberlain to a Premier League rival earlier in the day as the England international completed a move to Liverpool.
Kieran Gibbs was sold to West Brom on Wednesday while the futures of players such as Lucas Perez, Jack Wilshere and Mathieu Debuchy remain up in the air.
Leicester reportedly made a big-money move for Calum Chambers but Arsenal are adamant the defender will not leave in this window.
Source : PA