The midfielder was known to Leeds head coach at the time Uwe Rosler from his time in Norway, and the club's pursuit of the star continued through the month of June.
Leeds interest ended when Swiss giants Basel entered the race, and won Bjarnason over with the promise of Champions League football.
Signed at a bargain fee of just £1.5 million, Bjarnason has had an outstanding season, scoring 13 goals from midfield for Basel.
His efforts, which also included five assists, saw him land up with a Swiss league winner's medal, with Basel clinching the title by 14 points.
Bjarnason's goalscoring threat from midfield would have been a big asset for Leeds last season, who had just one player score into double figures, and no midfielder netting more than five.
Now he has a chance to show his talents on a big stage, with Iceland surprise qualifiers for Euro 2016. Unsurprisingly he played a prominent part in their qualification.
The 28-year-old is looking to add to his 47 international caps, in a group where they stand a fair chance of qualification.
Portugal are the favourites to top Group F, but Bjarnason's Iceland will fancy their chances amongst Austria and Hungary, and he has a chance to seriously enhance his reputation in the process.
England's Danny Rose (L) and Iceland's Birkir Bjarnason in action in 2011