The Cork-born player leaves Celtic having scored 29 goals in 76 appearances across two spells in Glasgow, first while on loan in 2023-24 and then after joining for a reported fee of £9.5m last summer.
Idah moved to Scotland from Norwich City, where he scored 17 goals in 116 appearances – 39 of which were starts – after coming through the club’s academy.
“We have wanted to add to Alan’s forward options during this window and a lot of hard work has gone in to trying to find a player who was the right person and the right fit for that role,” said Swansea director of football Richard Montague.
“Adam has a wealth of senior experience, including at the highest level of European club competition, and we expect him to make a considerable impact for Swansea City.”
Idah will become a team-mate of Zan Vipotnik at Swansea after the club ruled out selling the Slovenia international.
Vipotnik had been tipped to move on if Idah arrived, but is to remain in Wales having produced the best form of his Swansea career in the early part of the season.