The 23-year-old attacking midfielder is having a solid season in the Bundesliga, where he has scored seven times and laid on four assists for his team-mates.
He has three more years left to run on his contract, but it is becoming increasingly likely that he will be on his way out of Eintracht Frankfurt this summer.
Arsenal are showing strong interest in him and want to take him to the Emirates, but there are three more Premier League clubs who are also chasing the Dane.
According to German broadcaster Sport1, the Gunners are joined by three other Premier League sides in the chase, as Eintracht Frankfurt prepare to sell.
The German club are aware that they are unlikely to convince him to ignore the lure of playing in the Premier League next season.
Eintracht Frankurt are expected to ask for a fee in the region of €30m before giving their consent to his departure.
With big Premier League clubs in the race, the German club are confident of getting that figure for Lindstrom.
The midfielder is also a full Denmark international and nine international caps to his name.