Tore has caught the eye with some impressive performances for his club in recent times, form which has led to increased speculation surrounding the future of the winger.
The former Chelsea youth man has emerged as a primary transfer target of West Ham as Slaven Bilic looks to add more quality and depth to his squad ahead of an all-out assault on the top four next season.
A move to the Olympic Stadium had looked nailed on, with both the player and Bilic having confirmed their desire for the switch to materialise and the Besiktas president revealing that a bid had been made.
However, with the two clubs allegedly yet to agree a deal, Sabah report that Southampton have entered the fray as they attempt to railroad the seemingly imminent move.
According to the paper, the Saints have come in with an offer of €7 million (£5.8 million), but that figure is some way below the widely reported sum of €13 million (£10.8 million) that Besiktas are hoping to receive.
That is not the only reason that Southampton look less likely than the Hammers to complete a deal, with the London club also having the advantage of Bilic having coached Tore at Besiktas for two years.
There is also the fact that the Saints are currently managerless after Ronald Koeman departed for Everton, which means that Tore does not have the security of knowing the man he will be playing for, in stark contrast to the familiarity afforded by Biic.
Although the St. Mary's club finished one place above the Hammers last season, with both sides in Europe it would seem highly likely that Tore will lean towards reuniting with Bilic and joining West Ham's ambitious project if a move does indeed happen this summer.