The Monaco winger was heavily linked with a move to the Emirates in the last transfer window.
However, a potential £90m transfer fell through on deadline day.
The Gunners were expected to launch another attempt to sign the Frenchman, however according to the Daily Mirror, they are ready to look at alternative options.
Lemar has been linked with a host of European heavyweights in recent weeks, including Liverpool and Barcelona.
And Arsenal now believe Monaco will demand too much for one of the few remaining high-profile stars left at the club after their title-winning campaign last season.