The Belgium international is out of contract after five years at Old Trafford and free to negotiate with interested clubs.
According to The Sun, a proposed move to AC Milan has fallen through due to Fellaini's wage demands and now his representatives are looking closer to home for his next team.
The report states that new Gunners boss Unai Emery tried to sign Fellaini while at Paris Saint-Germain and is currently in the market for a tall central midfield player.
However, the Arsenal board reportedly believes that 30-year-old does not fit the club's style of play and is also seen as too expensive due to purported salary demands of £120,000 per week and a signing-on fee.
Fellaini, who made his name at Everton, is currently away with Belgium's World Cup squad and is not expected to make any decisions over his future until he returns from Russia next month.