If he accepts, Menezes, who has lifted two of the country's most popular clubs out of the doldrums, will have the huge task of trying to rebuild Brazil's national team and win a sixth world title when they host the 2014 World Cup.
Winning their own World Cup is seen as an obligation by Brazil's 190 million inhabitants.
The job, one of the toughest in international football, has been vacant since Dunga quit following the World Cup quarter-final defeat to Netherlands three weeks ago.
Ramalho, who has considerable domestic success but like Menezes is little known outside Brazil, sat down for talks on Friday with Brazilian Football Confederation (CBF) president Ricardo Teixeira, who praised the gruff 54-year-old's track record.
But hours later, Fluminense said that they wanted Ramalho to see out his contract, which runs until 2012.
"Muricy is going to continue at Fluminense, fulfilling his contractual commitments," club president Roberto Horcades told reporters.
In another twist, the CBF then said in a statement that Menezes had been approached and gave the impression his appointment was a mere formality.
"Mano Menezes will announce his decision at a news conference organised by his club Corinthians, tomorrow (Saturday), in Sao Paulo," said the statement on the CBF's website.
"Mano Menezes was one of a shortlist of three coaches and had his name ratified after a conversation with Ricardo Teixeira, when he showed himself to be in tune with the rebuilding project drawn up by the CBF for the 2014 World Cup."
Menezes, 48, made his name in 2005 when he led former South American champions Gremio out of the second division.
In an extraordinary decisive game, Gremio had four players sent off, survived a penalty miss by opponents Nautico and then snatched a goal to win 1-0 and clinch promotion.
Two years later, he took them to the final of the Libertadores Cup, the South American equivalent of the Champions League.
The following year he led Corinthians, another hugely popular team, out of the second division.
Former Brazil, Portugal and Chelsea coach Luiz Felipe Scolari, currently with Brazilian side Palmeiras, was believed to be the other coach on the shortlist.
Fluminense's move is almost unprecedented as Brazilian clubs are usually happy to oblige if their coach is picked for the national team.
Ramalho took over at Fluminense in April and on Thursday night his side beat Cruzeiro 1-0 to go top of the Brazilian championship.
Club president Horcades praised Ramalho for not leaving.
"People with Muricy's standards are necessary in football," he said.
The down-to-earth Ramalho won three consecutive Brazilian championship titles with Sao Paulo between 2006 and 2008, although he was never able to win the Libertadores.
Follow FFT.com on Twitter
Join FFT.com on
