Kane's office also sought to ensure that Williams would limit his review to the facts in the case and not focus on her decision not to bring charges.
The Inquirer has reported that after Kane took office last year, she shut down the sting. She has said she did so because she believed it was poorly run and possibly tainted by racial profiling, rendering it "not prosecutable." Those caught on tape were African American.
Abbott said Monday that Kane's concerns had been resolved and that she had notified Williams' office that the material would be forthcoming.
Williams, said Abbott, "made it clear he will review the case for prosecution and that he does not believe he has a conflict."
Williams' office could not be reached for immediate comment.
Several political and legal analysts have said they believe Kane faces a risk in handing the case over to the city District Attorney's Office, where the prosecutors who ran the state investigation now work.
A successful prosecution by Williams, or a scenario in which the district attorney makes public dramatic audio and video of legislators taking money, could be politically embarrassing for Kane's administration, they said.
Since The Inquirer first broke the news of the aborted investigation last month, Williams has been one of the biggest critics of Kane's decision to shut it down. She, in turn, has challenged Williams to take on the case, to see what he could do with it.
The sting began in 2010 under Republican Tom Corbett, then the attorney general and now governor. Over two years, it captured four state legislators and a Traffic Court judge accepting cash or, in one case, an expensive bracelet from an undercover operative, lobbyist Tyron B. Ali.
Over the last few weeks, Williams and Kane have exchanged a series of sharply worded letters about the case in an unusually public fight between two top prosecutors.
Last week, it appeared that Kane was backing down from handing over the case file. Williams dismissed Kane's suggestion of a conflict, countering that political endorsements do not grant immunity to those who give them.