The section of the Leveson report (you can read the executive summary here and the full report here) relating to "off-the-record" briefings has been interpreted by many journalists as a call for them to be banned. Such a measure would be both unjust and unenforceable but, thankfully, the report doesn't actually propose it.
First, the section in question relates specifically to contacts betweeen the press and the police, not those between the press and politicians (or anyone else). Second, it doesn't call for off-the-record briefings to banned, but for them to be described henceforth as "non-reportable" briefings. Leveson's objection isn't to the practice itself (he states that "everybody agrees that such briefings can operate in the public interest") but rather to the ambiguity created by the term "off-the-record".
Point 75 on p. 43 of the executive summary states:
The term 'off-the-record briefing' should be discontinued. The term 'non-reportable briefing' should be used to cover a background briefing which is not to be reported, and the term 'embargoed briefing' should be used to cover a situation where the content of the briefing may be reported but not until a specified event or time. These terms more neutrally describe what are legitimate police and media interactions.
Now, one could argue that renaming "off-the-record" briefings as "non-reportable" briefings is pointless semantics, but no one should claim that Leveson has called for the practice to be banned.