The coach has been in charge for 20 games after taking over from Paul Thompson.
Club owner Tony Smith regards the line-up as "Thommo's team" as the former coach put most of it together, but has been thankful for the new culture, tweaks and additions that have happened to the squad since the American arrived.
While results have been a mixed bag - Sheffield are floundering in the bottom half of the League and have slipped out of the Challenge Cup - Barrasso appears to be the man Smith would like to retain.Â Barrasso once intimated he was interested in a possible position in the Russian KHL.
"Talks are ongoing," said the owner. "Tom certainly has the respect of the guys that are working for him.
'It is no secret he has needed to improveÂ conditioning in the team and done that.
"He is very strong on the bench and has got the ability to get the best out of his players. We hope it turns out to be a long termÂ gig for Tom.
"We are a little bit thin on the ground in top quality bodies, I think it is fair to say.Â Our kids (younger players) are not quite ready yet and injuries to Davey Phillips and Eric Neiley have harmed us.
"Tom has been worried about playing with five 'D' - tiredness and potential injury is a threat to us, right now."
Steelers gave a debut to Sutton Sting part-time professional Jonas Fredriksson on Wednesday and while Smith says he does not know whether the Swede is "on the same level" as some of the other players, he identifies with the need to support the blue line.
"He has been training with us for a while, is looking for someone to give him a break, and I can see why Tom wanted to bring another body in, back there."
The proprietor said he'd "love to bring in another British defenceman" but none are available."
Steelers play Guildford Flames in the League at Sheffield Arena on Saturday night, with an eventual aim of getting on a run which brings them closer to the likes of leaders Belfast Giants, Cardiff Devils and Nottingham Panthers.Â