The Hammers began their interest in Meireles during last season's January transfer window, but were knocked back by the Turkish champions who were reluctant to sell.
Now, if reports are to be believed, an offer of around €4m will be enough to secure the services of the 31-year-old as Fenerbahçe look to balance their books.
Sam Allardyce will require a sizeable influx of players this summer in order to bolster his squad for next season.
In particular, West Ham need to strengthen the ranks in their midfield department following the release of Joe Cole and Jack Collison amongst others.
Meireles carries with him a wealth of experience including two season's worth of football in the Premier League.
The midfielder enjoyed short spells at both Liverpool and Chelsea, winning PFA Fans Player of the Year with the former and a Champions League title with the latter before departing for Turkey in 2012.
However the 31-year-old does not come without his baggage, Meireles is renowned for his controversial on-field manner.
Whilst at Fenerbahçe Meireles was handed an 11 match suspension for allegedly spitting and making offensive hand gestures at a Turkish referee. Later the same season Meireles was handed a further four match suspension for making inappropriate gestures towards a Galatasaray player during the Intercontinental derby.
West Ham are yet to confirm their interest in any possible deal for the Portuguese international but following their original interest just six months ago, it is likely that a cut-price deal for the midfielder could tempt Allardyce.
Meireles is currently on international duty in Brazil as the Portuguese prepare for a crucial game against USA in group G on the 22nd of June.