The reforms were approved in the parliament's lower house with a narrow majority of 22 votes for, 19 against and one abstention.
Nikola Spiric, the Bosnian prime minister, said: "I don't want to talk about dates, but I'm almost certain that we have fulfilled the last precondition set by Brussels."
The parliament's upper house is expected to approve the laws on Monday.
Earlier, the EU had insisted that the laws be passed for Bosnia to sign the stabilisation and association agreement, the first step towards membership.
But Serb opposition to the loss of what it saw as a cornerstone of its autonomy could mean that the merger of the two police forces will be largely cosmetic.
Under the new reforms, which were overseen by Miroslav Lajcak, Bosnia's international administrator, state bodies will control and co-ordinate the work of various agencies performing some of police activities at the state level, but will not interfere with the work of regional police forces.
The state control over local police agencies will be launched only after Bosnia agrees on the future organisation of the country in accordance to a new constitution.
The EU had pledged to accept whatever reform local politicians agree on as long as it met some broad criteria of operating in an unbiased manner.