The Kenyan midfielder – who was also being tracked by Newcastle, Birmingham, Aston Villa and Rangers – has agreed personal terms with the Parkhead club.
Germinal Beerschot, who have the 20-year-old under contract for another year, are willing to sell for £900,000 but, as Kenya are outwith the top 70 countries in the Fifa rankings, a work permit application is sure to fail.
The case will automatically go to appeal, which will be heard by a three-man committee brought in by the SFA. Celtic will have to convince the appeal panel that Wanyama is of sufficient quality to merit a permit.
But, as he has been playing international football for almost five years, they believe they have a strong case.
Wanyama has been on Neil Lennon's radar for 12 months and said: "I am extremely excited. Celtic is a legendary club. I'm looking forward to this.
"I have had a great time with Germinal Beerschot, and I look forward to the next chapter of my career at Celtic."
Although he is now in Australia for the club's pre-season tour, Lennon is being kept right up to date on developments regarding transfer activity in and out.
He has not given up hope Newcastle can be persuaded to sell Fraser Forster. But a previous target for the No.1 position, Real Valladolid keeper Jacobo, hopes he might be considered a back-up option.
The Spanish shot stopper played in a bounce game at Lennoxtown last summer and said: "I was heartbroken. I thought I would be a Celtic player, but the deal fell through.
"I hope something can be done with Celtic."
Bournemouth's 19-year-old strike starlet, Danny Ings, has also been linked with a move to Parkhead.
Meanwhile, Emilio Izaguirre has moved to put an end to speculation he wants to leave Parkhead. The Honduran defender – who has been linked with several clubs – said: "It is true there were offers from Manchester United, and other clubs.
"But I don't think anything will happen. My hope is to stay at Celtic. My aim is to recover my full fitness quickly so I can play again with my club."