The 23-year-old, who rose through the Gunners academy, will see his current deal expire in the summer.
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Gametime has been limited for Nelson this season, with the winger playing just 526 minutes of action for the club.
It was revealed in April that Arsenal had offered him a new contract in the hopes of extending his stay in north London.
However, Fabrizio Romano reports that Nelson has rejected their proposal, although negotiations are ongoing.
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A final decision on his future is said to be made at the end of the season, with a number of clubs in England and abroad keen on a free transfer should he leave.
Brighton, Aston Villa, Fulham and West Ham have previously been named as interested sides, as well as Champions League semi-finalists AC Milan.
Nelson is also said to be keen to stay at Arsenal, for whom he has made 63 senior appearances across five seasons.
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He made his first competitive appearance in the Gunners' Community Shield win against Chelsea in the 2017/18 season.
Nelson featured a further 15 times before heading out on loan to Hoffenheim in the following campaign.
The winger had a spell at Feyenoord last season and returned to the Arsenal first team in the summer.
Most of his appearances this season have come off the bench, one of which saw him score a stoppage-time winner against Bournemouth.
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Nelson has featured little since his heroic moment against the Cherries though and could be on his way out this summer.
There is unlikely to be a shortage of interest should he depart, especially with his three goals and three assists in 15 games this season.