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It is believed that another London club are also in the market for the 24-year-old, who is open to a move away from Brendan Rodgers' side.
According to football.london, the increasingly likely potential of Martin Odegaard fighting for his place at Real Madrid means that Mikel Arteta, who is an admirer of both Odegaard and Maddison, has switched his attention to the Englishman.
Leicester appear to be demanding a fee of £60 million for Maddison, which Arsenal are reluctant to pay due to budget issues.
They have, however, come up with an alternative offer believed to be a player plus money deal. It is currently unknown who the player is.
Arsenal do have a number of players who are available for a transfer this summer, including the likes of Sead Kolasinac, Ainsley Maitland-Niles, Joe Willock, Reiss Nelson and Eddie Nketiah.
Granit Xhaka, who looked set to move to Roma to play under Jose Mourinho, has signed a new long-term deal with the North London club.
Arsenal have already completed the signings of Nuno Tavares and Albert Sambi Lokonga this summer, but will be looking to further strengthen in order to challenge the top six once again.
Arteta's side had a poor last season, failing to qualify for the Europa League, meaning they are without European football for the first time in more than 20 years.
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