Danish FA wants Olsen to stay on
Olsen's contract runs until the end of June's finals in Poland and Ukraine and general secretary Jim Stjerne Hansen will ask the 62-year-old to keep going after 11 years in the job.
"I think that Morten Olsen is the coach with Danish blood in his veins who is best for this job," he told the country's biggest sports news website, sporten.dk.
"His results are indisputable, and the way the team plays bears witness to the fact that we are on the right path. But it takes two to tango."
Michael Krohn-Dehli and Nicklas Bendtner scored on Tuesday as Denmark beat Portugal 2-1 at home to clinch top spot in a tough Group H and make Euro 2012.
Olsen, who has now led the Danes to the finals of four major championships, was cautious about the future.
"Let's enjoy the fact that we're going to the Euros - the FA can take care of the rest," he told Danish television.
The widely-respected coach has recently been linked in the media with the vacant manager's job at Hamburg SV, a role currently being filled by fellow Dane and club sporting director Frank Arnesen.