The actor stars as Michael Kerrigan - a troubled priest who was abused as a child and is now trying to minister to a deprived northern community.
One viewer wrote on the social media website Twitter: "Seriously, just give the BAFTA to Sean Bean NOW... he has been magnificent in this."
Raw, exposing, emotional #broken simultaneously breaks hearts and brings the slightest degree of hope in all darkness #BAFTA #seanbean
— siobhan clarke (@sioclark) July 4, 2017
Another said: "Sean Bean has been immense in #Broken. Every week he has delivered a acting masterclass that's been compelling to watch. Worthy of awards."
Bean stars alongside Anna Friel - who plays mother-of-three Christina Fitzsimmons - in the BBC series created by screenwriter Jimmy McGovern.
Friel spoke highly of the programme, saying: "Just watched last episode of #BROKEN. Jimmy, Sean, mark, Ashley and Joel, everybody. Made me cry from beginning to end. Your talents amaze!"
"You wonderful #seanbean" the end was so emotional ❤ we're gonna miss u Father Michael #Broken #BAFTA @Ashleymoorex @LAProductionsUK
— Reem #SHQUAD (@reemy_filan) July 4, 2017
Another Twitter user said: "Every member of the cast and crew deserves an award for their work in this drama. This drama must get a Bafta."
Another said: "#Broken best drama this year. If Sean Bean doesn't get a BAFTA there's no justice in the world".
While, yet another said: "A BAFTA is definitely heading Sean Beans way, his acting in #Broken has been outstanding, there's SO much emotion in his performance."