Both teams are weighing up offers for the forward, whose future is uncertain, with the Saints signing Charlie Austin at the weekend.
Long is aiming to play regular football in order to stay sharp for Republic of Ireland at Euro 2016.
The 28-year-old was reported last week by the Irish Mirror to be a transfer target for both Liverpool and Tottenham Hotspur.
Long has scored five Premier League goals this season in 14 appearances, nine of which have been starts.
It is his second season at the club, last season netting five goals in 32 games, 16 starts, also providing five assists.
The Mirror reported this week that Southampton had rejected an £8 million offer from Liverpool for his services, and are determined to keep the striker.
Everton have been reliant on goals from Romelu Lukaku this season, and have money to spend after selling Stephen Naismith to Norwich, while Stoke are yet to see any of their goalscorers reach double figures this season,