Reports claim Bilic, who used to manage Fenerbache's rivals Besiktas, is keen on Sow who only has a year left to run on his contract and faces new competition from former Manchester United star Robin van Persie.
With Mauro Zarate and Modibo Maiga failing to impress, Carlton Cole having left and the injury-prone Andy Carroll out until at least September, West Ham are left with just Sakho and Enner Valencia up front.
Signing a striker is the club's number one priority before the start of the season. And the Hammers have been linked with a host of forwards this summer including Queens Park Rangers' Charlie Austin, Chelsea's Loic Remy and Manchester United's Javier Hernandez.
But with no sign of any movement on potential deals for those players, Bilic looks to be moving on to other targets.
West Ham to reap what they Sow
If the former Croatia boss does manage to get Sow, he will quite literally be buying goals.
The former France youth international has scored 82 goals in just 164 games for Fenerbache and before that Lille.
In fact during his time with Lille he fired in 25 Ligue 1 goals to lead his side to the title.
A final day hat-trick against his former club Stade Rennes sealed the league crown and also persuaded Fenerbache to splash out £7 million to bring him to Turkey.
Known for some acrobatic goals, including his trademark bicycle kicks, he actually scored a last gasp goal in a 1-0 win over Bilic's Besiktas side on his debut and has clearly left an impression on the Hammers manager.
West Ham fans should be excited at the potential partnership between Sakho and Sow who would be amongst the most feared pairings in the Premier League.