However, these links are now thought to be wide of the mark, according to the Bristol Post.
The Robins left-back is highly rated, and scored seven goals in 48 appearances last season for the Ashton Gate outfit.
The defender was also linked with Sean Dyche’s Burnley in May.
Bielsa’s side were rumoured to be in for Bryan, as he looks to strengthen his Leeds defence before the new season begins.
Bryan scores and assists regularly for a defender, so would have likely fit Bielsa’s attacking style he plans to bring in at Leeds.
However, it looks like this rumour is now well wide of the mark.
Leeds fans will have to wait a little longer to see what defensive reinforcements the Argentine will bring to Elland Road in this window.
Fans of the red half of Bristol will be relieved to see it looks like they are keeping hold of their man, for now.
However, Bryan has just one year remaining on his contract, so more rumours are likely to emerge surrounding the defender’s future.
If he does not sign a new contract with the Robins, they are in danger of losing the man for free at the end of the Season.
Leeds fans, what do you think of missing out on this target? Let us know in the comments below!