The Gunners are expected to make additions to their squad over the course of the transfer window and are closing in on Chelsea winger Willian.
They could opt to sell midfielder Lucas Torreira, who has been linked with a return to Serie A, and may look in the same league to secure a replacement.
The Premier League side have tested the water over a swoop for Diawara, according to Italian outlet Tutto Mercato.
It is unclear what answer Arsenal received and whether they were given a price at which they could sign Diawara.
The midfielder was praised for his performances under coach Paulo Fonseca this season and Roma could want to keep hold of him.
Diawara, 23, made a total of 22 appearances in Serie A for Roma over the course of the campaign.
The Italian giants have him locked down on a contract running through until the summer of 2024.