The Toffees have turned their attention to Iwobi after failing in their attempts to prise Wilfried Zaha away from Crystal Palace.
The Nigeria international may find game time harder to come by at the Emirates Stadium this season, with Arsenal having spent £72m on former Lille winger Nicolas Pepe.
However, the Gunners have turned down an opening offer of £25m up front, plus £5m in potential add-ons.
As per others, Everton £30m bid for Alex Iwobi rejected. Made to Arsenal today & knocked back. Expected to return with improved offer but likely need £40m+ to be accepted. 23yo not agitating but would go if deal agreed. #EFC interest recent & is in addition to Wilfied Zaha #CPFC— David Ornstein (@bbcsport_david) August 7, 2019
Everton, who are determined to strengthen in the wide positions before the transfer window closes on Thursday, are expected to return with an improved bid.
Iwobi made 51 appearances in all competitions last season, scoring six goals.
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