According to John Richardson of the Sunday Mirror, the Gunners will renew their interest in Lemar "in January or next summer."
However, Lemar's representatives also know of interest from Jose Mourinho's side, who will make a "club-record bid" for the AS Monaco winger.
The 21-year-old played a key role in Monaco's Ligue 1 title success last season, and WhoScored.com neatly summarised the attacking edge he provides:
Thomas Lemar: Only Thomas Mangani (15) has created more clear-cut goalscoring chances than Lemar (12) in Ligue 1 in 2017 https://t.co/VruRCQgrDp2017-9-6 14:00:04
Arsenal were keen to bring him to the Emirates Stadium at the end of the transfer window and submitted a £90 million bid, per Simon Mullock at the Mirror.
However, the move failed to materialise, and Gunners boss Arsene Wenger said it is "not true" Lemar rejected Arsenal but that "he decided to stay at Monaco," according to Pat McLoughlin at Sky Sports.
Wenger also stated Arsenal will try to sign Lemar again, although they may have to improve on last season's fifth-placed finish if they are to tempt the youngster to the Emirates:
B/R Football @brfootball
BREAKING: Arsene Wenger confirms that Arsenal will be back in for Monaco’s Thomas Lemar after failing to sign him this summer https://t.co/mEJJgn39hk2017-9-3 09:21:08
Meanwhile, the Gunners reportedly remain keen on Sterling, who is "understood to be open to a return to London," according to Simon Jones at the Mail on Sunday.
City boss Pep Guardiola refused to allow Sterling to leave for Arsenal in a straight swap with Alexis Sanchez on deadline day and wants the former Liverpool man to remain at the club.
However, City are expected to return with a £20 million bid for Sanchez in January, with Arsenal still keen on Sterling and Lemar as replacements, per Jones.
Guardiola was recently asked about the possibility of Sterling leaving and said there was "zero chance" and that the winger is "100 per cent committed" to Manchester City, per BBC Sport.
However, the City boss does have a whole host of attacking options to choose from, with Gabriel Jesus particularly impressive, per Opta:
13 - Gabriel Jesus has been directly involved in 13 goals in his 12 Premier League starts for Man City (nine goals, four assists). Menace. https://t.co/M1LMFlTZAE2017-9-9 12:23:01
Leroy Sane also made an impression on Saturday, arriving as a substitute in the 5-0 win over Liverpool and scoring twice, as WhoScored.com noted:
Leroy Sané: Has scored two goals in a Premier League match for the first time. He was only on the pitch 34 minutes #MCILIV2017-9-9 13:29:47
Sanchez's potential arrival in January would heighten the fierce competition for places, which may affect Sterling, who will want to be playing regular football ahead of the 2018 FIFA World Cup.