The Everton manager has already signed two defenders this summer, with centre-back Michael Keane arriving from Burnley for £25m and right-back Cuco Martina joining on a free transfer from Southampton, but wants a left sided defender.
Vermaelen, who can play both as centre-back and as left-back, has been identified as the player to fill the gap and the Toffees are hopeful a deal for the Belgian international can be reached before the end of the transfer window.
According to the Liverpool Echo, while Everton only want the 31-year-old on loan this campaign, their offer includes the option of making the switch permanent at the end of the season.
The Barcelona defender, who is also reportedly on Crystal Palace's radar, would add considerable experience to Everton's back four and would not need time to settle in the Premier League, having spent five seasons at Arsenal.
With the Gunners, the Belgian won the FA Cup in 2014 only a few months before moving to the Camp Nou for £17.1m. Vermaelen's spell in Catalonia, however, has been hampered by injuries and he featured in only 21 games across all competitions in his first two seasons at the club, before being shipped out on loan to Roma last summer.
Injuries continued to blight the Belgian, however, and he started only two Serie A games last season, making a meagre 12 appearances in total in all competitions.
Everton have spent over £120m on eight new signings so far this summer and are expected to sign Hajduk Split forward Nikola Vlasic, after the Croatian underwent a medical on Merseyside on Wednesday (30 August).
However, Koeman is keen to add a physical targetman to his ranks at Goodison Park after losing Romelu Lukaku to Manchester United but is still searching for the Belgium international's replacement with just over a day of the transfer window remaining.
Everton were monitoring the likes of Olivier Giroud, Edin Dzeko and Christian Benteke earlier this summer, but their options are now thin on the ground with such little time left to conduct their business.