The winger’s contract is set to expire at the end of this season.
Arsenal have opened talks with him over a new deal and offered to increase his £35,0000-a-week wages,
However, the Mail claim he could reject the proposal in the hunt for more regular first-team action.
West Ham are among the teams interested in Nelson with the player settled in his home city of London.
The likes of Fulham, Aston Villa and Brighton are also keeping tabs on the 23-year-old’s situation.
Villa boss Unai Emery is supposedly keen to work with Nelson again after managing him at the Gunners.
There is also interest coming from Europe via Serie A giants AC Milan.
Nelson has played 12 times in all competitions this season for Arsenal after battling a thigh issue that ruled him out of 17 games this campaign.
He has netted three goals and grabbed three assists with his best moment coming when he scored a last-gasp winner in a 3-2 victory over Bournemouth.
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Nelson spent last season out on loan at Feyenoord and helped them reach the Conference League final.
Rumours suggest that if he was to leave the club then Arsenal could look to replace him with Crystal Palace ace Wilfried Zaha.
The Ivorian is out of contract this summer and has also been linked with Borussia Dortmund, Monaco and Saudi Arabian clubs Al-Nassr and Al-Ittihad.