The Irons have targeted the Chelsea duo but are only expected to sign one of the forwards.
The Hammers are 11/8 to land Batshuayi while Giroud is a bigger price at 5/2 to move to the London Stadium.
But West Ham face a battle with Real Betis for Batshuayi, as the La Liga side are just 3/1 to swoop in and land the forward.
The East London club also enquired about the availability of Giroud following Gonzalo Higuain's arrival.
But the Blues are expected to cling on to the Frenchman after Alvaro Morata completed his switch to Atletico Madrid.
Betfair's Katie Baylis said: “Batshuayi hasn’t played for weeks for Valencia who he’s been on loan to and reports this afternoon looks like he could be going elsewhere for the rest of the season with West Ham 11/8 and Real Betis at 3/1.
"Everton have also been mentioned in relation to the Belgian and they are at 6/1."
She added: "“Reports out today say that West Ham may be considering a last minute bid for Giroud and we make it 5/2 they’ll nab him today.
"However, any move for the Frenchman would be more likely at the end of the season we think and we make it 1/4 he’ll still be at Chelsea tomorrow.”
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