Bilic confirmed on Thursday that he is still exploring options in the January transfer window after completing the signing of defender Sam Byram from Leeds United on Wednesday.
Injuries to Diafra Sakho and Andy Carroll, coupled with the potential departures of Nikica Jelavic and Mauro Zarate, has led to rumours of a move for Sturridge, despite the England international battling his own fitness problems.
Bilic revealed his admiration for the former Chelsea man, but has dismissed talk of a possible transfer, saying: "That is only a rumour and there is nothing behind it.
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When pressed on whether he could put forward an offer if Sturridge should become available, Bilic replied: "What if? I would take Lionel Messi also, but it is only an 'if' so there is nothing in it."
Bilic also confirmed both Carroll and Sakho are "ahead of schedule" in their efforts to recover from injury.
West Ham take on Manchester City at Upton Park on Saturday.