The 27-year-old joined the Toffees from relegated Aston Villa at the end of last season and has quickly established himself as a key member of Ronald Koeman's top-six-chasing team.
Everton would be reluctant to sell the defensive midfielder, who has been compared to PFA player of the year favourite N'Golo Kante, but The Mirror reports that Arsenal plan to test the Toffees' resolve.
The club are working on a number of transfer targets despite the ongoing uncertainty surrounding the future of manager Arsene Wenger, who is yet to publicly commit his future to the club beyond the end of the campaign.
Gueye has reportedly been earmarked as the ideal player to shore up Arsenal's defensive frailties, but the Gunners could face competition for his signature.
The Senegal international is understood to have caught the eye of Manchester United boss Jose Mourinho, while Sevilla, Valencia, RB Leipzig and Bayer Leverkusen have all also been credited with an interest.