The Mail reports that the Gunners have made enquiries about signing the 21-year-old this month.
Southampton have shown interest in signing their former academy product but his £140k a month salary could prove problematic.
Everton and Southampton have both been linked with bids for the England global in recent weeks but according to the Daily Mail, the Toffees are now "confident" that they will beat their rivals to his signature.
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Jordan Henderson and Alberto Moreno were both absent as they continue their respective rehabs while Nathaniel Clyne is still AWOL.
Walcott would face a hard decision if Arsenal were to accept a bid, as he has a young family in the south, so may choose Southampton and take a pay-cut to join the Saints.
Niasse has struggled for first-team football at Goodison Park and at one stage did not even have a locker at the Merseyside club under former boss Ronald Koeman.
Walcott's directness would add a lot to Southampton's game, while he managed to score 19 goals in 37 games across all competitions last season.
The Chilean is expected to leave Arsenal in the summer, although the Gunners could allow Alexis to leave the club during the January transfer window if they are able to recoup most of the £38m they paid to Barcelona in 2014.