Adams surprised the Magpies when he requested a move in the days after their premiership.
The 30-year-old is seeking more midfield time at the Swans, having spent stints at half-forward with Collingwood this year following the arrival of on-baller Tom Mitchell.
Adams will reunite with former Magpies teammate Brodie Grundy at Sydney, who had been plotting a move for the ruckman during the back half of the season.
But Beatson said Adams' availability only became known "very late" in the finals series.
The 206-game veteran missed Collingwood's grand final win over the Brisbane Lions with a hamstring injury sustained at training two weeks earlier.
"If I had to hazard a guess, probably somewhere between the preliminary final and the grand final, when he got his injury," Beatson told SEN radio of the transfer timeline.
"We were approached by his manager (Winston Rous from Phoenix Management Group), would it be something we would look at?
"We were looking to bolster our midfield with an experienced and good stoppage player with some power.
"It was very late and it grew from there pretty quickly."
Adams' trade was confirmed on Wednesday, with the Swans securing their target in a straight swap for pick 33 in this year's draft.
Grundy cost the Swans this year's pick 46 and a future second-round selection.
His move to a third club in as many years came after Melbourne's experiment of pairing Grundy with captain Max Gawn failed.
Beatson said the "jungle drums were beating" midway through the season around Grundy being unhappy with limited ruck opportunities.
The 29-year-old will slot straight into the Swans' No.1 ruck role vacated by the retired Tom Hickey.
Beatson expects Grundy to relish the relative anonymity that comes with being an AFL footballer in Sydney.
"The thought of a new environment where he's not going to be hassled every two minutes was something that he's looking forward to," Beatson said.
"He can walk around the streets of Sydney and not be pulled up every two seconds for a selfie.
"Some people thrive on that, others relish the chance to get away from it. For Brodie, that will be a plus."
Beatson confirmed Sydney "had a crack" at recruiting Aaron Naughton before the key forward signed a long-term contract with the Western Bulldogs.
He also said the Swans aren't concerned about losing emerging tall Logan McDonald, a West Australian product who has reportedly attracted interest from his home state.
"He's contracted for next year. He seems really close with the player group," Beatson said.
"We've already expressed our interest to his manager that we want to extend him and we'll be working on that at the start of the year.
"All indications are that he's really tight with that young group and they see a future together. We're comfortable with that at the moment."
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