The move follows Lord Justice Leveson's damning report into the phone hacking scandal.
Both the First Minister and David Cameron have warned the media that change is needed.
But Mr Salmond has also said that he backs a new Scottish regulator based on the Irish system.
However, that model has been rejected by the Prime Minister because it includes so-called "statutory underpinning".
Mr Cameron warned that any use of legislation could lead to politicians meddling in the freedom of the press.
The disagreement has opened up the possibility of different regulators in Scotland and the rest of the UK.
The decision lies with the Scottish Government as, while broadcasting is reserved to Westminster, regulation of the press is devolved.
But at the weekend all three main opposition leaders in Scotland questioned the wisdom of having a separate Scottish system. Critics have also queried the practicality of such a move.
They warn of the difficulties of stopping newspapers crossing the Border and question how papers' internet sites would be regulated by two different regimes.
The Prime Minister's official spokesman said that as well as cross-party talks with Labour and the LibDems, he expected cross-governmental talks as well.
"I'm sure that will form part of the communications between the Department of Culture, Media and Sport and others," he said.
A Scottish Government spokesman said that ministers would "be happy to have talks".
Scottish parties are due to meet for talks later this week.
But opposition leaders will use the occasion to ask Mr Salmond to step aside from the process.
It follows the findings in the Leveson report that Mr Salmond had shown a "striking" willingness to lobby for Rupert Murdoch's proposed £8 billion takeover of BSkyB.
But the Leveson report also said that the SNP leader could not be criticised because he had not actually lobbied any ministers over the issue.