This is according to reports from the player's native Denmark cited by The Daily Star, which suggest that the Gunners are aiming to beat Danish Superliga outfit Brondby - Agger's former club - to the centre-back's signature this summer.
The same source further reports that any Arsenal move for Agger would arise as an attempt to secure a replacement for in-demand captain Thomas Vermaelen.
Belgian international defender Vermaelen is widely expected to leave the club during the current transfer window, with rumours linking him with both Manchester United and Barcelona continuing to gather pace amid claims that the player does not want to enter negotiations over a new contract at the Emirates Stadium.
Such a stance leaves Arsene Wenger in somewhat of a predicament. While he presumably won't be particularly pleased about the prospect of losing a player with Vermaelen's versatility, leadership and experience, he will be well aware that the former Ajax man's current deal is set to expire in a year's time and to keep hold of him would then mean running a significant risk of losing him for free in 2015.
Like Vermaelen, Agger is also expected to leave his current club in the coming weeks having found regular starts somewhat difficult to come by in recent times.
Although the 29-year-old made 20 appearances in the Premier League for Liverpool last season, he was rarely trusted by manager Brendan Rodgers to perform on the biggest occasions.
With Agger already struggling to start ahead of the likes of Martin Skrtel and Mamadou Sakho, the recent purchase of Croatian centre-back Dejan Lovren from Southampton is likely to hasten his exit from Anfield.
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Arsenal and Brondby are not the only clubs credited with an interest in pursuing Agger, however.
Barcelona - who have been very active in the market during the early months of Luis Enrique's managerial reign at the Camp Nou - have been persistently linked with a move of their own for Agger at a cost of approximately £12 million.
With the La Liga giants having somewhat neglected their defence in recent years, Enrique appears keen to ensure that Barcelona will not be left too disadvantaged by the retirement of club stalwart and former skipper Carles Puyol.
Blaugrana have already completed the signing of one senior defender in recent weeks, with Frenchman Jeremy Mathieu arriving from rivals Valencia.