Adiyiah, 20, was top scorer when Ghana won the Under-20 World Cup in October and is being bought because Milan's pickup in form has coincided with a switch to a 4-3-3 formation.
"The deal with the Norwegian club is done, now he has to get permission to come to Italy. Naturally he must pass a medical," Galliani told Milan's television channel.
"Our tactics have changed, we needed a wide forward who can alternate with those already here and we've made our move."
Los Angeles Galaxy midfielder Beckham spent five months on loan at Milan last season and a repeat spell from January has been in the pipeline for months as he aims to secure a place in England's World Cup squad.
"In a matter of hours, I hope to be able to announce the return of David Beckham," added Galliani, who is celebrating 30 years working for Milan owner Silvio Berlusconi.
Despite his joy at the signings and Milan's 2-0 win over Parma on Saturday, Galliani was worried by another low attendance figure at the San Siro.
"It's a shame that there's been an unjustified drop-off in the attendance. We accept it even if the figures are worrying. We'll try everything possible to recover our fans," he said.
Leonardo's side had a poor start to the season but are now up to fourth in Serie A.
