He travelled to Glasgow on Monday to discuss personal terms and undergo a medical with the Hoops.
Rasmussen, who has three caps for his country, was recently in Thailand on duty with a Danish league select.
Rasmussen, who is nicknamed Duncan due to a resemblance to former Rangers striker Duncan Ferguson, reportedly cost Celtic more than £1million.
He started his career with AGF Aarhus before moving in 2006 to Brondby, where he scored 52 goals in 111 games.
He has also played for his country at every age group from Under-16 onwards.
While many Hoops fans believe the current strike partnership of Georgios Samaras and Marc-Antoine Fortune spend too much of their time outside the opposition penalty area, Rasmussen promised he is exactly the opposite.
Speaking to the media at Celtic's Lennoxtown training complex ahead of the SPL clash with Hibernian at Parkhead, where he is likely to feature at some stage, he said: "It is a big thing for me, it's a dream.
"Celtic are a big club so I'm pleased to be here.
"It's a very big jump up from playing with Brondby.
"I'm a striker and most of my goals are scored in the box.
"I am a box player, I like to be in the box."
Tony Mowbray believes he has added a "different type of striker" to his options.
He said: "I am delighted that we could add to the options up front.
"He's a different type of striker to the ones that we have got.
"Hopefully, given the opportunity, he can score goals which will obviously help the team.
"Samaras and Fortune do a lot of their very best work outside the box and working defences very hard.
"Scott McDonald has a more predatory instinct and gets lots of goals with his twisting and turning but this boy can score different types of goals and he is a clinical finisher given the opportunity.
"What we like is desire to score and the enjoyment when he does score."
Celtic v Hibernian To Score 3+ Goals: Celtic 11/8