The 27-year-old spent last season on loan with Real Madrid and remains likely to leave Old Trafford despite being one of just three recognised strikers at Louis van Gaal's disposal.
Enner Valencia, who picked up an injury in the Europa League 2-2 draw with Astra Giurgiu, is expected to be out for up to 10 weeks, forcing Bilic's hand in the transfer market.
According to the London Evening Standard, the East London side have already failed with one £8.5m offer earlier in the summer but, with Diafra Sakho being the only forward available, are likely to return with an improved package.
West Ham remain interested in luring Charlie Austin from Championship side Queens Park Rangers but are not prepared to pay £15m to sign the England striker.
Meanwhile, the Hammers have agreed a season-long loan deal for Atletico Madrid striker Raul Jimenez but now face a battle to obtain a work permit for the Mexico striker.
Teddy Sheringham, the former United striker, has also urged his past employers to sign a new striker if they are to seriously challenge for the Premier League title this season.
"They’re making an awful lot of signings and there are still bits going on," he told 888Sport.
"I would presume there is still a striker coming in for Man United and I’d be very surprised if that doesn’t happen. That could make all the difference."