If passed into law by a second and third reading it would alter current practice under which at the end of an enquiry police tell the attorney general if they feel that they have enough evidence for a prosecution.
Netanyahu has so far been questioned six times by detectives of the police national fraud and serious crimes squad over two suspected cases of corruption.
The bill is being fast-tracked by the government and was rushed through a parliamentary preparatory committee hours before the house voted for it by 46 votes to 37.
It said the first would allow the attorney general to ask the police for "input in the existing corruption probes into Prime Minister Benjamin Netanyahu."
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But the police opinion could not be made public, it wrote.
"The second revision sets a one-year prison sentence for investigators who leak their conclusions to outside sources."
Since the police started grilling Netanyahu in January, Israeli media have been full of leaks on the probe's progress, some reportedly coming from within the police force itself.
The bill's sponsor, David Amsalem of Netanyahu's Likud party, said he sought only to protect the rights and reputation of suspects.
"The situation today is that the attorney general can make the (police) recommendations public," it quoted Dov Khenin of the opposition Joint List as saying.
"According to the new clause, he cannot. This is a dramatic and extreme move. You are silencing the attorney general."
Netanyahu is suspected of having received luxury gifts from wealthy supporters, including Israeli businessman and Hollywood producer Arnon Milchan, who has also been questioned.