In a surprise intervention, the First Minister said Mr Dyke would restore public confidence in the wake of the corporation's botched Newsnight report wrongly linking allegations of child sex abuse with a senior Tory.
The controversy forced George Entwistle to resign on Saturday after just two months as director-general. A permanent successor has yet to be named.
Mr Salmond said: "The crisis engulfing the BBC underlines serious and institutional failures of leadership that must immediately be addressed.
"The BBC ship needs to be steadied by someone who the public can have confidence in to return the corporation to its journalistic traditions.
"The obvious figure who fits the bill is Greg Dyke."
Mr Dyke stepped down in 2004, along with then-chairman Gavyn Davies, after Lord Hutton's damning report on the David Kelly affair and BBC journalism in the run-up to the Iraq war.
Lord Hutton said a BBC report claiming the Government "sexed up" a dossier on Iraq's weapons of mass destruction was "unfounded", though no WMDs were discovered after the conflict. The whistleblower, scientist David Kelly, committed suicide.
Mr Salmond, who wants to replace the BBC with a Scottish broadcasting service if Scotland becomes independent, added: "To a great extent, journalism at the BBC has suffered since the day Greg Dyke resigned following the Hutton Inquiry.
"It is clear now from everything that has been revealed post the Iraq war that Greg Dyke was right to defend the BBC journalists at that time.
"The crisis currently engulfing the BBC must not be allowed to obscure the underlying issues relating to extremely serious allegations of abuse and the exploitation of children which, although they may date back a number of years, must still be investigated.
"Greg Dyke is someone the public would also have confidence and trust in when it comes to taking forward the corporation's own inquiries into that issue."
The SNP appealed to Mr Dyke a year before his resignation to support a Scottish Six TV news bulletin, over the UK-wide Six O'Clock News north of the Border.
Senior Nationalists believed he was more sympathetic than his predecessor, John Birt.
In an apparent reference to the move, Mr Salmond said Mr Dyke "fully understood the obligation on the corporation to have the resources to be able to act as a public service broadcaster for the nation of Scotland".
He added: "Reappointing Mr Dyke would restore confidence in BBC journalism.
"No doubt he would be a difficult appointment for some in the establishment to swallow, but he is certainly a man the public would have confidence in – and public confidence is the prime requirement for a public service broadcaster."
Mr Dyke was a popular director-general whose resignation sparked walkouts by disappointed staff in BBC centres, including Glasgow.
Mr Entwistle has been replaced temporarily by Tim Davie, the corporation's director of audio and music.
The bookies' favourite to land the job permanently is Ofcom boss Ed Richards.
Mr Richards, 45, is former senior adviser to Tony Blair. As a former head of corporate strategy at the BBC, he was linked to the job prior to Mr Entwistle.
However, he was described by Mr Dyke as a "jumped-up Millbank oik", in a reference to his role working for former the Prime Minister, during a 2003 Royal Television Society conference.
Mr Davie is another bookies' favourites to head the corporation. Caroline Thomson, the former BBC chief operating officer, last night ruled herself out. She will today be named as the two-day-a-week chairwoman of Digital UK, the body responsible for ensuring people connect to digital services, on a salary of £50,000.
Ms Thomson reportedly described herself as "not a candidate" as she was "wanting to get on with my other career".
The BBC has suspended a string of senior executives pending further inquiries into the Newsnight show on November 2.
The investigation, run in conjunction with the Bureau of Investigative Journalism, mistakenly connected a Tory peer in the North Wales sex abuse scandal. He was named on social media as Lord McAlpine, but it became clear the case was based on mistaken identity.
On Friday, he categorically denied the allegations, his accuser Steve Messham retracted his claim and the BBC was forced to apologise.
l The BBC was too slow to deal with the unfolding Jimmy Savile sex scandal, according to its former director of Global News Richard Sambrook.
The former member of the corporation's management board said it was "very slow to spot the toxic nature of the story".
Dame Janet Smith, who is conducting an inquiry into the Savile claims, has launched a witness appeal and a website was set up for alleged victims.
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