The Gunners have already seen their £30 million bid rejected by Ligue 1 champions Monaco but quickly returned with a £40m offer.
Les Rouges et Blancs are yet to respond but the expectation is it will be rejected once more, report football.london.
But Arsene Wenger will not be deterred and will return with a third bid in order to get his man to follow record signing Alexandre Lacazette.
Monaco are said to be determined to hold on to the 21-year-old, having already lost Bernardo Silva to Manchester City this summer.
But interestingly the French club have left Lemar out of the promotional picture for their new away shirt which has just been released.
Lemar was in the home kit poster which emerged towards the end of last month, along with Arsenal target Kylian Mbappe.
Mbappe takes centre stage one again in the new picture for the away kit – but enticingly for Gunners fans Lemar is conspicuous by his absence.
Wenger is determined to land the France international, who he almost signed when he was still at Caen in 2015, but he instead opted to move to the Stade Louis II.
The winger scored 14 goals last season to help Monaco to the Ligue 1 title.
Arsenal meanwhile have opened talks with Leicester over Riyad Mahrez, who they see as a fall-back option should they not sign Lemar.
Discussions have been held with the Foxes for the Algeria international according to Telegraph Sport, but the two clubs are some way off agreeing a fee.
Arsenal value the 26-year-old at around £35m but Leicester are holding out for closer to £50m for the 2016 PFA player of the season.
Mahrez has handed in a transfer request in a bid to force a move away and is keen on a switch to the Emirates. He is due to report back for pre-season training at Leicester today.