The Shadow Culture Secretary is also calling for a "robust fit and proper person" test to be introduced under a fresh push for curbs on the industry.
Lord Justice Leveson failed to make much progress on suggestions that market share should be limited during the inquiry into media ethics sparked by the phone hacking scandal, she will say. The Labour frontbencher will call on her Tory counterpart, Maria Miller, to set up cross-party talks to help push reforms through.
Labour last year proposed a possible 30% limit on ownership within the newspaper industry but suggests that could sit alongside a 15% cap across the media as whole, including "any medium of communication that stands between a creator of content and an audience".
News International held 37% of the newspaper market until the closure of the News of the World.
Speaking at the annual Charles Wheeler Lecture on journalism, hosted by the University of Westminster and the British Journalism Review, she will say: "The Leveson Inquiry focused on the complaints system and the impunity which came from the lack of a robust, effective complaints system was undoubtedly a key part of the problem.
She will say: "But we don't have a proper regime for protecting against this. The system doesn't work - its inadequacies and complexities were laid bare by the News Corp bid for the whole of BSkyB. And the system is out of date - this is an age of great change in the media, where we have print newspapers, broadcast media and new media, and a convergence of all three."
Ms Harman will say there should be a cross-party process, which should also involve the industry, the outcome of which could be used to come up with proposals that form part of a new Communications Act.
"There are a number of key decisions that the process should take," she will say. "It should decide on the best way of measuring media power. It should decide on the proper upper limit for cross-media ownership." She will add: "It should decide on a proper upper limit for media ownership within sectors. Although it is important that our rules on cross media ownership are changed, the rules on how much of the newspaper market (for example) one company is allowed to own must also be changed."